One of the arguments I have heard that he did not have a proof is the following.

He wrote his note in his copy of Arithmetica around 1637.

He most likely wrote his proof for the case of n=4 in the 1640s.

He sent letters to other mathematicians in 1640, 1657 where he talks about the case of n=3 but writes in such a way that it seems like he does not have the answer.

Why would he write n=4 after? If he had a generalized proof?

Why would he tease other mathematicians with a special case in 1657 if he already had a generalized proof?