Most of the remaining air leaked out of Tom Brady’s attempt to overturn his “Deflategate” suspension on Wednesday when a federal appeals court rejected his request for a new hearing.

The Patriots quarterback had hoped the full United States Court of Appeals for the Second Circuit in Manhattan would hear his case, after a three-judge panel ruled by a 2-1 vote in April that NFL commissioner Roger Goodell was within his rights when he suspended Brady for four games.

Instead the request by Brady and the NFL Players Association was flatly dismissed in a brief ruling that said, “IT IS HEREBY ORDERED that the petition is denied.”

So barring a seemingly highly unlikely successful appeal to the U.S. Supreme Court — or a change of heart by Goodell — Brady will miss the Patriots’ first four games, starting Sept. 11 against the Cardinals in Glendale, Arizona. The Patriots then play the Dolphins, Texans and Bills, all at home. Backup Jimmy Garoppolo would be in line to start in Brady’s place.

The Deflategate saga dates to Brady’s alleged role in, or at least knowledge of, a plan to make footballs easier to handle by deflating them under the legal standard for the AFC Championship Game against the Colts on Jan. 18, 2015.

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Brady was allowed to play the entire 2015 season as the case made its way through legal channels. But Wednesday’s decision confirmed that Goodell has extensive powers under the NFL’s collective bargaining agreement.

The NFL Players Association took issue with the ruling, saying in a statement: “We are disappointed with the decision denying a rehearing, as there were clear violations of our collective bargaining agreement by the NFL and commissioner Roger Goodell.

“ . . . The track record of this league office when it comes to matters of player discipline is bad for our business and bad for our game. We have a broken system that must be fixed.

“We will review all of our options carefully on behalf of Tom Brady and all NFL players.”

League investigator Ted Wells initially determined the Patriots had intentionally underinflated footballs and that Brady was “at least generally aware” of the plan. Goodell reacted by suspending Brady for four games and fining the team $1 million and taking away two draft picks.

Brady was able to keep playing when U.S. District Judge Richard Berman overturned the suspension last Sept. 3, allowing Brady to start last season’s opener. But the league appealed to the second circuit, and the three-judge panel reinstated the ban. Brady sought either a rehearing by that panel or a hearing by the entire court and got neither.

The drawn-out controversy has driven a wedge between the NFL and its most successful franchise of the 21st century, as well as between the league and Patriots fans, most of whom have remained fiercely loyal to their four-time Super Bowl-winning quarterback. It also highlighted the extreme power Goodell has over disciplinary matters, a subject that is certain to be a matter of debate the next time the NFL must negotiate a collective bargaining agreement with its players.

Assuming Brady’s suspension finally sticks, or at least remains unresolved before the Sept. 11 opener, the Patriots must find a way to juggle getting Brady ready during training camp while also preparing Garoppolo to start. Garoppolo has completed 20 of 31 passes, one for a touchdown, in 11 games in his regular-season career.

The other quarterback on the roster is rookie Jacoby Brissett. So the Patriots might bring in a veteran backup for the first month of the season — assuming Brady finally stays sidelined.