Let $A$ be a finite dimensional $K$-algebra for $K$ a field and let $I(A)$ denote the injective envelope of the regular module $A_A$. Define the stable $A$-module $\underline{A}$ as the cokernel of the trace of $I(A)$ in $A$. Recall here that the trace of a module $N$ in a module $M$ is the maximal submodule of $M$ generated by $N$, see for example https://www.math.uni-bielefeld.de/~ringel/opus/good-d-r.pdf at the end of page 2.

I have the guess that for a general $A$-module $M$ we have the formula $$pdim M= sup \{ i \geq 0 \mid Ext_A^i(M,\underline{A}) \neq 0 \}.$$

Question: Is this true?

This would have some nice homological applications.