The comments by <a href="https://mathoverflow.net/questions/357317/results-that-are-widely-accepted-but-no-proof-has-appeared#comment898311_357317">Monroe Eskew</a> and <a href="https://mathoverflow.net/questions/357317/results-that-are-widely-accepted-but-no-proof-has-appeared#comment898408_357317">Andrés E. Caicedo</a> concerning unpublished results of Hugh Woodin deserve to be made into an answer IMO. As a concrete example, Caicedo wrote:

> There is the fact that Turing determinacy implies Suslin-coSuslin determinacy (in the presence of DC?), which gives L(R)-determinacy.

There are various other results by Woodin that may or may not fit the bill; in many (though maybe not all) cases, proofs have been provided by other authors.  For more details, see <a href="https://mathoverflow.net/q/138308">Woodin's unpublished proof of the global failure of GCH</a> and <a href="https://mathoverflow.net/q/132326">Unpublished works of Woodin on SCH and Radin forcing</a>.