Let $R$ be a local artinian Gorenstein ring and $M$ a finitely generated $R$module, then $Ext_R^1(M,M) = 0$ if only if $M$ is projective?
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This would be a very strong version of the AuslanderReiten Conjecture (see here, for example) in the Gorenstein case. The Conjecture is still open, though many partial results are known. By the way, this paper (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". 

