$Ext_{A^e}^i(D(A),A)$ for finite dimensional algebras

Question

Let $A$ be a finite dimensional non-semisimple algebra over a field $K$ with enveloping algebra $A^e=A^{op} \otimes_K A$. Let $D(A)=Hom_K(A,K)$.

Question:

Is there always a positive integer $i>0$ with $Ext_{A^e}^i(D(A),A) \neq 0$?

Even for hereditary algebras $A$ (where it should be true), I have not found a good approach to this. I have a proof for hereditary algebras of Dynkin type and some other cases.

See also Secret exact sequence in path algebras of Dynkin type