JUDGMENT

These appeals from judgments of the United States District Court for the District of Columbia were presented to the court, and briefed and argued by counsel. The court has afforded the issues full consideration and has determined that they do not warrant a published opinion. See D.C. CIR. R. 36(d). For the reasons stated in the accompanying memorandum, it is

ORDERED and ADJUDGED that the judgment of conviction and sentence with respect to appellant Eddie Ray Kahn and the judgments of conviction with respect to appellants Stephen C. Hunter, Allan J. Tanguay, and Danny True be affirmed. It is

FURTHER ORDERED and ADJUDGED that the sentences of appellants Stephen C. Hunter, Allan J. Tanguay, and Danny True be vacated and that the cases be remanded to the district court for re-sentencing.

Pursuant to D.C. CIR. R. 36, this disposition will not be published. The Clerk is directed to withhold issuance of the mandate herein until seven days after resolution of any timely petition for rehearing or rehearing en banc. See Fed. R.App. P. 41(b); D.C.Cir. R. 41(a)(1).

MEMORANDUM

I.

Eddie Ray Kahn, Stephen C. Hunter, Allan J. Tanguay, and Danny True appeal from their convictions and sentences on charges of conspiracy to defraud the United States and mail fraud, in violation of 18 U.S.C. §§ 371, 1341, and 2. According to the superseding indictment against the four defendants, Kahn was the founder and leader of American Rights Litigators (ARL), an organization that promoted and sold tax defiance schemes, and Hunter, Tanguay, and True were ARL employees. Between 1996 and 2004, ARL enrolled more than 4,000 customers in all fifty states, the District of Columbia, and several foreign countries. ARL sought to help its customers frustrate the tax collection and enforcement efforts of the Internal Revenue Service (IRS) through myriad schemes.

The indictment highlights two specific schemes. In one scheme, known as “the UCC Process,” ARL sold fraudulent “Bills of Exchange” to its customers, who then submitted those fictitious financial instruments to the IRS in purported payment of their federal tax debts. In a second scheme, ARL and its customers inundated the Treasury Inspector General for Tax Administration with frivolous complaints against IRS officials who worked on cases involving ARL members. ARL’s operations ceased after federal law enforcement officials executed a search warrant at the organization’s offices in early 2004.

In September 2008, Kahn, Hunter, Tan-guay, True, and a fifth co-defendant, Jerry R. Williamson, were indicted in federal district court in the District of Columbia. Williamson pled guilty in April 2009, and a *7superseding indictment against the remaining four defendants followed in April 2010. A 21-day trial commenced later that month, and the jury returned a guilty verdict against Hunter, Kahn, Tanguay, and True on all counts charged in the superseding indictment. The district court sentenced Kahn to concurrent terms of imprisonment of 240 months for mail fraud and 60 months for conspiracy. The court sentenced Hunter, Tanguay, and True each to concurrent terms of 120 months on the mail fraud charges and 60 months for conspiracy — sentences that the court indicated were well below the applicable guideline range. The court also fined each of the defendants $25,000 and ordered three years of supervised release.

II.

On appeal, Kahn, who is not an attorney, states that he is “appearing personally” but “not pro se.” Kahn maintains that he “was not the defendant in this case,” that the United States of America is an “improper plaintiff,” that the district court judges who presided over the case are executive branch officers, that the Justice Department attorneys who handled his case lacked the authority to prosecute him because they are not members of the judicial branch, and that the indictment against him was invalid because — among other alleged infirmities — it cited the mail fraud and conspiracy statutes by their U.S.Code sections rather than referencing specific acts of Congress. Those claims and similar arguments in Kahn’s appellate briefs are “so lacking in merit that they d[o] not deserve any further comment.” See Billman v. Comm’r, 847 F.2d 887, 889 (D.C.Cir. 1988) (per curiam) (alterations and internal quotation marks omitted).

Hunter, Tanguay, and True (the “joint appellants”) have filed joint submissions on appeal. The joint appellants’ principal argument for reversal of their convictions is that Congress intended for crimes like theirs to be prosecuted under the criminal provisions of the tax code — in particular, 26 U.S.C. §§ 7206 and 7212 — rather than the mail fraud statute. As the district court correctly concluded, however, that argument “cannot be sustained because it runs aground of controlling precedent.” Mem. & Order 1, Oct. 6, 2009, ECF No. 168. See United States v. Dale, 991 F.2d 819, 849 (D.C.Cir. 1993) (per curiam) (holding that “the tax code is not the exclusive regime under which tax fraud schemes may be prosecuted”).

The joint appellants also challenge the district court’s denial of True’s motion to dismiss for discovery abuse, which accused the government of engaging in “ ‘dump truck’ discovery.” Mot. to Dismiss at 5, July 13, 2009, ECF No. 122. Again, we affirm for the reasons given by the district court. See Order, Oct. 6, 2009, ECF No. 165. The joint appellants acknowledge that they have found no criminal case in which a court has reversed a conviction based on an argument of this sort, and as the district court observed, the government appears to have exceeded its discovery obligations here. Id. at 2.

The joint appellants next argue that the district court improperly allowed testimony by a government expert witness, William C. Kerr, and improperly excluded testimony from True’s expert witness, Walker F. Todd. Kerr, a longtime national bank examiner at the Office of the Comptroller of the Currency and an expert on genuine and fictitious financial instruments, reviewed various “bills of exchange” created by ARL. He highlighted the characteristics of those documents that lent them a veneer of validity and the features that rendered them valueless. The joint appellants raise various challenges to Kerr’s testimony, none of which has merit. Be*8cause Kerr gave live testimony and was subject to cross-examination, his testimony did not violate the Confrontation Clause of the Sixth Amendment. Kerr’s testimony also did not intrude upon the province of the jury, did not give rise to a significant danger of unfair prejudice, and was potentially helpful to the jury in understanding the evidence. The district court acted well within its discretion in allowing Kerr to testify.

Meanwhile, True sought to introduce expert testimony from Todd, a former attorney at the Federal Reserve Banks of New York and Cleveland who has appeared as an expert witness in other district courts. True’s proffer stated that Todd would testify about the history of the U.S. monetary system, U.S. enforcement efforts related to fraudulent bills of exchange, the IRS’s ability to identify these fraudulent financial instruments, and the views of “Redemption Theory” adherents who supposedly believe that such bills of exchange are valid. The district court did not abuse its discretion in disallowing Todd’s testimony. As the court correctly noted, the monetary history of the United States was not at issue; the federal government’s enforcement efforts with respect to other suspects who used fraudulent bills of exchange do not bear on these defendants’ guilt or innocence; and True’s proffer failed to explain why Todd, who had never worked for the IRS, had any basis to testify concerning the IRS’s ability to identify fraudulent bills of exchange. Mem. & Order 1-5, May 11, 2010, ECF No. 295.

We also find that the district court committed no abuse of discretion when it excluded Todd’s testimony concerning “Redemption Theory.” See United States v. Hall, 969 F.2d 1102, 1109-10 (D.C.Cir. 1992). True sought to introduce that testimony to show that, “because of a misunderstanding of the law, he had a good-faith belief that he was not violating any of the provisions” of the relevant statutes. See Cheek v. United States, 498 U.S. 192, 202, 111 S.Ct. 604, 112 L.Ed.2d 617 (1991). Cheek allows defendants to pursue such a mistake-of-law argument for criminal tax offenses that require the mens rea of willfulness, and similar mistake-of-law arguments have been recognized under other criminal statutes that make willfulness an element of the offense. See Bryan v. United States, 524 U.S. 184, 191-92, 118 S.Ct. 1939, 141 L.Ed.2d 197 (1998) (“As a general matter ..., in order to establish a ‘willful’ violation of a statute, ‘the Government must prove that the defendant acted with knowledge that his conduct was unlawful.’” (quoting Ratzlaf v. United States, 510 U.S. 135, 137, 114 S.Ct. 655, 126 L.Ed.2d 615 (1994))); see also United States v. Moran, 493 F.3d 1002, 1013 (9th Cir. 2007) (“good faith belief in legality” is defense to mail fraud). A defendant’s views about the validity of the relevant statutes, however, are “irrelevant to the issue of willfulness.” Cheek, 498 U.S. at 206, 111 S.Ct. 604. Here, True’s proffer stated that Redemption Theory adherents “disagreed with the Government’s prevailing view regarding the nature of money and underpinnings of the federal tax system.” See True’s Notice of Intent to Offer Expert Test. 2, Apr. 5, 2010, ECF No. 253. That proffer failed to make clear whether Redemption Theory adherents believe that submitting “bills of exchange” to the IRS in purported payment of tax debts does not violate the law (which would be relevant to guilt or innocence) or only that federal tax laws should not apply to them (which would be of no relevance). True thus failed to carry his burden of showing that the expert testimony was admissible. See Meister v. Med. Eng’g Corp., 267 F.3d 1123, 1127 (D.C.Cir. 2001) (proponent of expert testimony bears the burden of showing its admissibility).

*9Moreover, while courts of appeals have suggested that defendants in tax-related cases may be allowed to introduce expert testimony in certain circumstances to establish that they had a good-faith belief in the lawfulness of their conduct, e.g., United States v. Harris, 942 F.2d 1125, 1132 n. 6 (7th Cir. 1991); United States v. Burton, 737 F.2d 439, 443 (5th Cir. 1984), expert testimony concerning legal materials on which a defendant did not rely is irrelevant because “only the defendant’s subjective belief is at issue.” Harris, 942 F.2d at 1132 n. 6; see also United States v. Daly, 756 F.2d 1076, 1083-84 (5th Cir. 1985) (expert testimony has “little probative value on the issue of the defendants’ states of mind at the time they acted” where there is “no evidence that [the defendants] had relied on [the expert’s] opinion at the time they acted”). Here, True’s proffer did not state that he actually held — or even was aware of — the “non-traditional views” about which Todd would testify. See True’s Notice of Intent to Offer Expert Test. 3, Apr. 5, 2010, ECF No. 253.

The joint appellants further argue that their convictions must be overturned on sufficiency-of-the-evidence grounds because the fraudulent bills of exchange at issue were “replete with disqualifying marks” and “gibberish.” But as our decisions make clear, a fraud is a fraud even if “no reasonable person would have believed [the defendant’s] misrepresentations.” United States v. Maxwell, 920 F.2d 1028, 1036 (D.C.Cir. 1990); see also United States v. Brien, 617 F.2d 299, 311 (1st Cir. 1980) (collecting cases). The joint appellants also contend that the district court erred by excluding summary evidence showing that Eddie Ray Kahn had received the bulk of the benefits from ARL. The district court, however, was well within its discretion to conclude that the attempt to introduce that evidence — on the fifteenth day of trial — simply came too late. In any event, the summary evidence was of doubtful relevance. While a defendant’s “stake in the venture” is “not irrelevant to the question of conspiracy,” Direct Sales Co. v. United States, 319 U.S. 703, 713, 63 S.Ct. 1265, 87 L.Ed. 1674 (1943) (internal quotation marks omitted), the government never argued that Hunter, Tanguay, and True received any portion of ARL’s profits; their “stakes in the venture,” according to the government, were the salaries they received from ARL. Finally, the joint appellants erroneously argue that the district court failed to instruct the jury with respect to their mistake-of-law defense. Cf. Cheek, 498 U.S. at 200-03, 111 S.Ct. 604. The instructions correctly stated that a “good-faith misunderstanding or belief’ as to what the law requires would constitute a defense to conspiracy and mail fraud.

III.

Hunter, Tanguay, and True also raise a series of sentencing-related challenges. First, the joint appellants argue that the district court should have calculated their base offense level according to the Sentencing Guidelines table for tax offenses, see U.S.S.G. § 2T4.1 (2009), rather than the Guidelines table applicable to fraud offenses, see U.S.S.G. § 2Bl.l(b). That argument is premised on the joint appellants’ claim that they should have been charged under the criminal tax statutes rather than the conspiracy and mail fraud statutes — a claim we have rejected.1

*10Second, the joint appellants challenge the district court’s finding at sentencing that they are responsible for an intended loss of $42.7 million, a figure that includes the aggregate value of the check-like bills of exchange found on ARL’s computers and that corresponds to a 22-level increase in their total offense level under the loss table for fraud offenses. See U.S.S.G. § 2Bl.l(b)(l)(L). The joint appellants argue alternatively that (1) they “knew [the bills of exchange] were worthless” and that (2) they thought the bills of exchange were legitimate. A party may, of course, make arguments in the alternative, see Nat'l Com Growers Ass’n v. EPA, 613 F.3d 266, 275 (D.C.Cir. 2010), but those diametrically opposite arguments are meritless. The “intended loss” figure may include losses that “would have been impossible or unlikely to occur,” U.S.S.G. § 2B1.1 application n. 3 (A)(ii); and to the extent that Hunter, Tanguay, and True argued that they had a good-faith belief in the validity of the bills of exchange, the jury rejected that argument in its verdict.

Third, the joint appellants argue that they were entitled to downward departures under U.S.S.G. § 2B1.1 application n. 19(C), which states that “a downward departure may be warranted” when the offense level calculated under § 2B1.1 “substantially overstates the seriousness of the offense.”2 While a district court has discretion to depart downwards in such circumstances, the joint appellants identify nothing in the record indicating that the district court misunderstood its authority to depart, and the district court ultimately granted significant downward variances to Hunter, Tanguay, and True at a later stage of the sentencing process. See United States v. Booker, 543 U.S. 220, 264-65, 125 S.Ct. 738, 160 L.Ed.2d 621 (2005); 18 U.S.C. § 3553(a). Assuming, arguendo, that a district court’s refusal to depart is reviewable where the court properly understood the scope of its discretion under the Guidelines, cf. United States v. Olivares, 473 F.3d 1224, 1230-1231 (D.C.Cir. 2006), we see no reason to conclude that the district court abused its discretion here.

Fourth, the joint appellants argue that the disparity between their ten-year sentences and the ten-month sentence for co-defendant Jerry Williamson renders their sentences “procedurally improper” and “substantively unreasonable.” Williamson, however, pled guilty to only one count of mail fraud based on two fraudulent bills of exchange — one for $7,120 that he tried to use to pay part of a fine in a prior criminal case, and one for $8,000 that he sent to the Bureau of Land Management. His convicted conduct is substantially less culpable than that of Hunter, Tanguay, and True.

Fifth, the joint appellants challenge the district court’s application of the official victim guideline, which provides for a three-level increase in a defendant’s offense level when the offense was motivated by the victim’s status as a government employee. See U.S.S.G. § 3A1.2(a). The three-level increase was based on evidence that ARL and its customers sent more than 1,400 complaints to the Treasury Inspector General for Tax Administration falsely accusing IRS employees of misconduct. In similar circumstances, two courts of appeals have determined that government employees are “victims” for purposes of U.S.S.G. § 3A1.2 when the defendant illegally files false claims of misconduct against them in an attempt to trigger in*11vestigation by the IRS. See United States v. Anderson, 353 F.3d 490, 497, 507 (6th Cir. 2003); United States v. Hildebrandt, 961 F.2d 116, 117, 119 (8th Cir. 1992). Aside from their unsuccessful attempt to disclaim responsibility for this aspect of the ARL conspiracy, the joint appellants give no reason to diverge from Anderson and Hildebrandt, and they appear to accept those holdings.

The joint appellants’ most meritorious argument — and the only one that requires us to remand — is that the district court improperly applied a three-level upward adjustment to their Guidelines offense level on the ground that they were “manager[s]” or “supervisor[s]” of a criminal activity that involved five or more “participants” or was “otherwise extensive.” See U.S.S.G. § 3Bl.l(b). To qualify for a three-level upward adjustment under § 3Bl.l(b), “the defendant must have been the ... manager ... or supervisor of one or more other participants,” and the Guidelines commentary clarifies that a person qualifies as a “participant” only if she is “criminally responsible for the commission of the offense” (although she “need not have been convicted”). See U.S.S.G. § 3B1.1 application nn. 1-2. The ARL conspiracy plainly was a criminal activity involving multiple criminally responsible “participants.” What is less clear is whether Hunter, Tanguay, or True supervised any other “participant” who was “criminally responsible.”

In its sentencing memorandum and its brief on appeal, the government argues that Hunter, Tanguay, and True each managed or supervised other ARL employees, including ARL’s receptionist and employees who sent letters to the IRS on behalf of the organization. But at the sentencing stage of the district court proceedings and in its written submissions to this court, the government never argued that any of those lower-level ARL employees were criminally responsible for the commission of any offense. The first time that the government suggested that any of those lower-level ARL employees were criminally responsible participants within the meaning of U.S.S.G. § 3B1.1 was at oral argument before this panel. Arguments raised for the first time at oral argument, however, are considered forfeited. United States v. Southerland, 486 F.3d 1355, 1360 (D.C.Cir. 2007).

We therefore conclude that the government has failed to carry its burden of showing that the three-level enhancement under U.S.S.G. § 3Bl.l(b) applies to Hunter, Tanguay, and True. See United States v. Bapack, 129 F.3d 1320, 1324 (D.C.Cir. 1997) (“At sentencing, it is the Government’s burden to demonstrate by a fair preponderance of the evidence that an enhancement is warranted.”). We also reject the government’s request — again, raised for the first time at oral argument — that we apply the harmless-error doctrine in these circumstances. Improper calculation of the guideline range is a “significant procedural error,” Gall v. United States, 552 U.S. 38, 51, 128 S.Ct. 586, 169 L.Ed.2d 445 (2007), and where, as here, the government has not timely presented a harmless-error argument, we are reluctant to raise and resolve the issue sua sponte. See United States v. Pryce, 938 F.2d 1343, 1347 (D.C.Cir. 1991) (opinion of Williams, J.). One of the factors that courts consider in deciding whether to embark upon a harmless-error inquiry on their own initiative is “whether a reversal will result in protracted, costly, and ultimately futile proceedings in the district court.” United States v. Giovannetti, 928 F.2d 225, 227 (7th Cir. 1991); accord United States v. Davis, 726 F.3d 434, 445 n. 8 (3d Cir. 2013); United States v. Mitchell, 502 F.3d 931, 977-78 (9th Cir. 2007). Here, there is no reason to believe that a remand for resen-*12tenting will lead to “protracted,” “costly,” or “futile” proceedings. See United States v. Blackson, 709 F.3d 36, 40 (D.C.Cir. 2013) (setting forth this circuit’s rules regarding the scope of the district court’s resentenc-ing authority under a remand order).