Let $A$ be an algebra with dominant dimension at least one and $Af$ a minimal faithful projective-injective left $A$-module (but we use right modules in the following). The Nakayama conjecture states that $Ext_A^i(A/AfA,A) \neq 0$ for some $i \geq 1$ (this is equivalent to finite dominant dimension), while the first Tachikawa conjecture states that $Ext_A^i(D(A),A) \neq 0$ for some $i \geq 1$. It is well known that the Nakayama conjecture implies the first Tachikawa conjecture. The other direct seems to be not known.

Now I noted the relation $\Omega^{-1}(\tau(A/AfA))=D(A)$.

Can this somehow (or some other trick) be used to show that $Ext_A^i(D(A),A) \neq 0$ for some $i \geq 1$ implies $Ext_A^i(A/AfA,A) \neq 0$ for some $i \geq 1$?