Not signing the extension is surely all the org needs to prove who is telling the truth. Both sides agreed it was offered. Speculation around the league was Fox was done with the Kings and the extension was proof. But there was a cope: Fox was merely holding out for the Designated Veteran Exception (the 35% of cap flavored super-max).
By my count 4 players have signed the DVE and gone on to win a championship after signing (Jaylen Brown, Steph Curry, Giannis and Nicola Jokic). Jason Tatum signed one immediately after winning a championship last year.
The others are: Russ, Harden, Wall, Dame, Gobert, Embiid, Booker, and KAT.
Players who refused to sign one include Kawhi and AD who immediately won chips with their next team.
Anyways, the track record for winning after signing this extension is not good. The Celtics already had a Finals appearance before Brown signed, Steph already had several rings, and Giannis and Nicola were widely regarded the two best players in the league. If Fox valued winning he would not have prioritized it. Of course if he had gone out and earned it last year we would have made the playoffs, Brown would be coach, and all of this would be moot.