Tom Brady’s high-powered attorneys filed a beautifully written and fairly persuasive appeal brief Monday, asking the US Court of Appeals for the Second Circuit to grant the Patriots quarterback an “en banc’’ rehearing to review last month’s 2-1 decision that reinstated Brady’s four-game suspension for Deflategate.

Yet Brady still faces a steep uphill climb to even get his appeal heard, let alone win it. Between 2000 and 2010, less than .03 percent of appeals in the Second Circuit received an en banc review, as the Second Circuit is notorious for respecting the decisions of its three-judge panels.

“Obviously, it is well done, as one would expect,’’ said Suffolk Law professor Marc Greenbaum of the brief, which was written by Ted Olson, Brady’s newest lead attorney and a former US solicitor general. “And even more obviously, it is still a long shot.’’

But most of Deflategate up until now has defied the odds — federal district judge Richard Berman took the rare step of vacating an arbitration award, and the Second Circuit took the rare step of reversing Berman — and Brady’s legal team certainly presented a convincing argument Monday that their appeal at least deserves to be heard by the entire Second Circuit.

“The divided panel of the Second Circuit reached erroneous legal conclusions under an unfair and unjust standard,’’ Olson said in a statement. “The decision and the standards it imposes are damaging and unfair — not only to Tom Brady but to all parties to collective bargaining agreements everywhere.’’

Brady’s legal team needs to persuade at least seven of the 13 active judges that further review is necessary. In order to achieve the statistically improbable, Brady’s lawyers need to convince the court that their case is of extraordinary circumstances and has wide-ranging implications beyond the quarterback’s suspension and the interpretation of commissioner Roger Goodell’s powers in the NFL’s collective bargaining agreement.

Not surprisingly, their brief played up this sentiment.

“The panel decision will harm not just NFL players, but all unionized workers who have bargained for appeal rights as a protection,’’ Olson wrote. “The panel decision will also harm management by freeing labor arbitrators from collectively bargained limitations on their authority, enabling them to dole out their own brand of industrial justice.

“A divided panel of this Court affirmed Goodell in a decision that repudiates long-standing labor law principles and that, if left undisturbed, will fuel unpredictability in labor arbitrations everywhere, and make labor arbitration increasingly capricious and undesirable for employers and employees alike.’’

Judges Barrington Parker and Denny Chin ruled in favor of the NFL in their decision last month, deciding that Article 46 of the CBA grants Goodell wide-ranging powers to penalize players who violate the integrity of the game, and that he was within his rights to hand Brady a four-game suspension.

Chief judge Robert Katzmann dissented, ruling that Goodell unfairly dispensed his own brand of “industrial justice.’’

Chin and Katzmann are among the 13 judges who will decide whether to grant Brady an en banc review.

While the timetable is hard to predict, a decision on whether to grant the review should come within four to six weeks. If the review is heard, then the suspension is temporarily stayed, and Brady could potentially play the entire 2016 season while the legal process runs its course.

Brady “was not afforded fundamental fairness and due process as guaranteed by the collective bargaining agreement and case law,’’ NFL Players Association executive director DeMaurice Smith said. “We also know that the NFL propped up a now completely debunked ‘independent’ report with a made-up standard as the basis for his suspension.

“For 60 years, we have affirmed the right to seek redress for our members, and we will always hold the NFL accountable.’’

Olson outlined two key areas in which he believes Parker and Chin erred and why Brady deserves an en banc review:

¦  Goodell was not permitted to affirm Brady’s suspension on new grounds at his NFL appeal hearing last June. Goodell affirmed the punishment after learning that Brady had destroyed his cellphone.

¦  Any ball-deflation scheme should be considered an equipment violation, and Goodell was wrong to not consider other punishments that were potentially applicable, such as a fine. Instead, Goodell compared Brady’s conduct to steroid use and applied a similar four-game suspension.

To bolster his first point, Olson cited Stolt-Nielsen S.A. v. AnimalFeeds International Corp., in which “the Supreme Court reaffirmed that an arbitration award must be vacated ‘when an arbitrator strays from interpretation and application of the agreement and effectively dispenses his own brand of industrial justice.’?’’

Greenbaum believes that this argument is the most compelling.

“A real arbitrator would have rejected the NFL’s attempt to shift the basis on which the original discipline was premised,’’ said Greenbaum, himself an arbitrator. “The key here is that the commissioner was supposed to hear an ‘appeal.’ What he did was akin to the following: A defendant appeals his or her conviction of a misdeameanor, and the appellate court finds that the defendant was really guilty of a felony. That is not supposed to happen in the United States.’’

The NFL’s attorneys will not have to respond to Olson’s brief unless requested by the Second Circuit, which could be seen as a sign that at least seven judges want to review this case.

Should Brady get his en banc review in front of 14 judges — the 13 active judges and Parker — he would need at least a 7-7 split to have Berman’s decision reinstated and his suspension vacated.

Olson and Team Brady did everything they could in Monday’s brief to convince the Second Circuit that Brady’s case deserves to be reviewed further, for the sake of all union workers.

“Under the panel majority’s misguided approach, an arbitrator is now free to ignore critical provisions of a CBA reflecting collectively bargained penalties,’’ Olson wrote. “The panel decision stands in stark conflict with fundamental rules of labor law and undermines the rights of union members and employers alike. This Court should grant rehearing.’’

Ben Volin can be reached at ben.volin@globe.com. Follow him on Twitter @BenVolin.