Let $A$ be a finite dimensional Hopf algebra over a field $K$ that is weakly symmetric (meaning $soc P = top P$ for each indecomposable projective $A$-module P).
Question: Is $A$ then automatically symmetric?
Here symmetric means $A \cong Hom_K(A,K)$ as $A$-bimodules.
A positive answer to this question would also give a positive answer to Are local finite dimensional Hopf algebras symmetric? .
