By Bryan Mena, CNN

Washington (CNN) — Federal Reserve Governor Lisa Cook has the support of every living former chair of the central bank’s powerful Board in her legal battle with President Donald Trump, who tried to fire her last month based on unproven allegations of mortgage fraud, according to an amicus brief filed to the Supreme Court Thursday.

An appeals court earlier this month kept Cook in her post through a preliminary injunction while her lawsuit challenging Trump’s firing attempt moves forward — just days before the central bank’s September policy meeting. The administration appealed that decision, and is now being considered by the nation’s highest court.

Former Fed chairs Alan Greenspan, Ben Bernanke and Janet Yellen warned against overturning the injunction, stating that it would “threaten” the Fed’s independence of politics and “erode public confidence in the Fed.” The brief was also signed off by some Republicans who once served in high-ranking government roles, such as former Treasury Secretary Henry Paulson and former Council of Economic Advisers Chair Glenn Hubbard.

The 32-page document details the historical importance of the Fed’s policy decisions being free of political considerations, which is mostly the role it plays in keeping the US economy stable. At stake is the Fed’s hard-fought reputation with investors and the rest of the world, according to the brief.

Damaging the Fed’s credibility could interfere with one of the institution’s core missions of keeping inflation in check, the brief said.

“The Fed’s ability to fight inflation is directly related not only to its actual insulation from short-term political pressures but also to the public’s perception of its independence—because if the public and financial markets believe that the Federal Reserve is sufficiently insulated, they will act in accordance with that expectation, resulting in lower and more stable inflation, which is consistent with lower long-term interest rates,” according to the brief.

Meanwhile, the allegations of mortgage fraud against Cook haven’t been taken to court. Federal Housing Finance Agency Director Bill Pulte first raised the accusations on social media and it is now being investigated by the Justice Department. So far, the administration hasn’t revealed a smoking gun.

Cook’s legal battle is part of Trump’s broader attempt to reshape the Fed’s top ranks.

At the central bank’s rate-setting meeting earlier this month, Cook deliberated the economy alongside the newest Fed Governor Stephen Miran, a top economic adviser to Trump and who is taking an unpaid leave to serve a vacated term on the central bank’s Board of Governors. Miran was sworn in an hour before the Fed’s September 16-17 meeting, he told CNBC last week.

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