Recall that a finite dimensional algebra $A$ over a field $K$is called a Frobenius algebra in case $A \cong D(A)$ as right modules, where $D(A) \cong Hom_K(A,K)$. In case $A \cong D(A)$ as bimodules, then $A$ is called a symmetric algebra.
It is well known that a finite dimensional Hopf algebra is a Frobenius algebra.

>Question: Is a finite dimensional local Hopf algebra a symmetric algebra?

Examples of local finite dimensional Hopf algebras are group algebras of $p$-groups and it was shown in https://mathoverflow.net/questions/255143/when-is-the-exterior-algebra-a-hopf-algebra that for the exterior algebra, it is indeed true that being a Hopf algebra implies that it is symmetric.