Given a finite dimensional local Hopf algebra $A$ over a field $K$ and two finite dimensional indecomposable modules $N$ and $M$. Is it known when the module $N^{*} \otimes_K M$ is projective? Can this happen when $M$ and $N$ are not projective? Can it happen for a fixed $M$ and $N=\Omega^k(M)$ for some $k$?

Special cases such as $A$ being $KG$ for a $p$-group $G$ are also welcome.