It looks like he may soon (finally) be heading to serve his sentence. The three-judge panel gave him until May 17, 2024 to petition for an en banc review before the entire US Court of Appeals for the District of Columbia, but this is unlikely to succeed. The next stop is evidently the Supreme Court, where Chief Justice Roberts will be the ultimate decision maker whether the highest court will hear this case.
As always, you can read the full decision here.