Let $A$ be a Nakayama algebra and $M$ be the direct sum of all indecomposable $A$-modules $N$ with $Ext_A^1(N,N) \neq 0$.

The following is suggested by computer experiments with QPA:

Question: Is $B=End_A(M)$ an Auslander-Solberg algebra?

Recall that an algebra $B$ is an Auslander-Solberg algebra in case we have $Gordim(B) \leq 2 \leq domdim(B)$ and they can be equivalently characterised as the algebras with the Gorenstein projective modules being an abelian category, see https://link.springer.com/article/10.1007/s10468-013-9448-5 .

In Combinatorial problem on periodic dyck paths from homological algebra it was proved that with $N$ indecomposable non-rigid, also $\Omega^1(N)$ has the same property. Thus we can associate to each Nakayama algebra a graph with points the indecomposable non-rigid modules and arrows $N \rightarrow \Omega^1(N)$. In case the above question has a positive answer, it might suggest that this graph has some nice properties. Besides that I have no real approch to this question since $M$ is not even a generator or cogenerator, which is the usual setting for Auslander-Solberg algebras.