Let $A$ be a finite dimensional Frobenius algebra and $e$ and idempotent of $A$. It is well known that the algebra $eAe$ does not have to be a Frobenius algebra. But if $A$ is additionally symmetric, then $eAe$ is also a symmetric Frobenius algebra for any idempotent $e$.

The Frobenius algebra $A$ is called weakly symmetric if for every indecomposable projective module $P$: $top(P)=soc(P)$.

Question: If $A$ is just weakly symmetric, is $eAe$ also always weakly-symmetric for any idempotent $e$?