I saw someone post a GPT-5.6 proof of the (3,2) conjecture here the other day, so I figured I’d drop this here too.
Two days ago, I asked GPT-5.6 Ultra to prove the Toeplitz square peg conjecture. After 3 hours of thinking, it said it couldn't complete the proof - but claimed to have cracked Terence Tao’s Conjecture 5.6 instead, from his 2016 preprint “An Integration Approach to the Toeplitz Square Peg Problem”.
I'm definitely not qualified to verify the math myself, but every AI model I asked to critique it said the proof looks solid. Claude Fable Max thought about it for about 20 minutes and couldn't find a mistake. It hasn't been reviewed by a human expert.
I’m posting it here mainly to get it out there, just in case it turns out to be correct. As far as I understand, it wouldn't be extremely significant since some weaker consequences have already been proved independently, but it is an open conjecture and Tao said about it: ~"Conjecture 5.6 appears difficult to resolve in general" and describes it as the ~"strongest of the new conjectures in the paper".
Would love to hear if any actual mathematicians can spot an error.