The Wedderburn-projective formulation of the Idempotent Nakayama Conjecture

Let $B$ be a Noetherian algebra, and let $N$ be a Wedderburn projective $B$-module. Write $\tau_N(B)$ for the trace of $N$ in $B$. Wedderburn-projective Idempotent Nakayama Conjecture. If

$$\operatorname{Ext}^{i}_{B}(B/\tau_N(B),B)=0\quad\text{for $i\ge 0$},$$

then $N$ is a generator. The source states that this formulation is equivalent to the Idempotent Nakayama Conjecture and notes that it appears weaker than the Strong Nakayama Conjecture because over many rings a Wedderburn projective module is automatically a generator.