Let $R$ be a local artinian Gorenstein ring and $M$ a finitely generated $R$-module, then
$\mathrm{Ext}\_R^1(M,M) = 0$ if only if $M$ is projective?



Selforthogonal modules over Artinian Gorenstein rings
This would be a very strong version of the Auslander-Reiten Conjecture ([see here](https://www.leuschke.org/research/AR-Conjecture/), for example) in the Gorenstein case. The Conjecture is still open, though many partial results are known.


By the way, [this paper](https://doi.org/10.1016/j.jalgebra.2008.04.027) (ScienceDirect link, may not be visible to everyone) claims a proof of the Conjecture in the Gorenstein case, which would give an affirmative answer to your question, but I --- and several other people I've talked to --- believe there is a gap in the proof. Their assumption is that $\mathrm{Ext}\_R^{i}(M,M)=0$ for $i =1,2$. The questionable step is at the top of page 2163, the second line, where they say "therefore ... is exact".


Edit 2023-05-15: The paper I referred to above is: "When are Torsionless modules projective?", by Rong Luo and Zhaoyong Huang,
Journal of Algebra 320 (2008) 2156-2164, MR2437647.


