A very difficult Diophantine problem $n^2 \mid 3^n+2^n+1$

Question

Prove that $n=3$ is the only positive integer greater than $1$, for which$$n^2 \mid 3^n+2^n+1.$$

This is a conjecture.

Answer 1

Worth an answer. It is fairly likely that there is no known solution for this. Several years ago we had some nonsense with the similar $$(n^2 - 1) | (3^n + 5^n)$$

See https://mathoverflow.net/questions/16341/on-polynomials-dividing-exponentials

I admit, the question here differs in a way that may be important, as the analogous MO problem would be $$n^2 | (1 + 3^n + 5^n)$$

The same question was asked here, Find all positive integers $n$ s.t. $3^n + 5^n$ is divisible by $n^2 - 1$ Gottfried put in a good deal of effort, however, if you check his comments below his answer, he realizes that he does not have a complete proof, although there are places inside the answer that claim completeness.