A finite dimensional (connected if needed) $K$-algebra $A$ over a field $K$ is called symmetric when $A \cong Hom_K(A,K)$ as $A$-bimodules.

Being symmetric implies that all terms of Hochschild cohomology and homology are isomorphic (as $k$-vector spaces). Is the other direction also true, that is:

Question: When all terms of Hochschild cohomology and homology are isomorphic for an algebra $A$, is $A$ symmetric? Is this at least true when $A$ is a Frobenius algebra?

If this is true, maybe one can use this to generalise the definition of symmetric algebras to more general rings.

A finite dimensional (connected if needed) $K$-algebra $A$ over a field $K$ is called symmetric when $A \cong Hom_K(A,K)$ as $A$-bimodules. Symmetric algebras are Frobenius algebras and include for example any group algebra of a finite group.

Being symmetric implies that all terms of Hochschild cohomology and homology are isomorphic (as $k$-vector spaces). Is the other direction also true, that is:

Question: When all terms of Hochschild cohomology and homology are isomorphic for an algebra $A$, is $A$ symmetric? Is this at least true when $A$ is a Frobenius algebra?

If this is true, maybe one can use this to generalise the definition of symmetric algebras to more general rings.