Tachikawa's first conjecture

Let $A$ be a finite-dimensional algebra over a field. The condition that all higher extensions from its $k$-dual back to $A$ vanish is

$$\operatorname{Ext}_A^i(D(A),A)=0 \quad\text{for all } i\geq 1.$$

Here $D(A)$ denotes the $k$-dual of $A$. Tachikawa's first conjecture. If

$$\operatorname{Ext}_A^i(D(A),A)=0 \quad\text{for all } i\geq 1,$$

then $A$ is selfinjective. This conjecture is a consequence of the Nakayama conjecture, which states that every non-selfinjective finite-dimensional algebra has finite dominant dimension.