Given a finite dimensional selfinjective algebra $A$ and an indecomposable module $M$, which is not projective. Let $v=DHom(-,A)$ be the Nakayama functor. In case $Ext^{i}(v(M),M) \neq 0$ for some i, can we say that $Ext^{j}(M,M) \neq 0$ for some $j$ in general?
This would have some consequences for the Nakayama conjecture, so special cases are welcome too.
