Let $A$ be a finite dimensional local Frobenius algebra with simple module $k$ and an indecomposable non-projective module $M$ (that is also finite dimensional).

Question:

Is there an example of such an $M$ such that every map $f: \Omega^n(M) \rightarrow k \rightarrow \Omega^{n-1}(M)$ is zero in the stable module category of $A$ for all $n \in \mathbb{Z}$?

Note that $\operatorname{Ext}_A^1(M,M) \cong \underline{\operatorname{Hom}_A}(\Omega^{n}(M),\Omega^{n-1}(M))$ and it seems to be unknown whether this can be zero.

$\operatorname{Ext}_A^1(M,M)$ is for example always non-zero for radical cube zero algebras or Hopf algebras.

(I reformulated the question as suggested by a comment of Jeremy Rickard)