Self-orthogonal quasi-generator conjecture

Let $A$ be a finite-dimensional algebra over a field $k$. For $n\in\mathbb{N}\cup\{0\}$, an $n$-quasi-generator is a module admitting the quasi-generator property of level $n$, and a module $T$ is self-orthogonal when $\operatorname{Ext}_A^{i>0}(T,T)=0$. Self-orthogonal quasi-generator conjecture. Every self-orthogonal $n$-quasi-generator has projective dimension at most $n$. If true, this would imply that every self-orthogonal $n$-quasi-generator is an $n$-tilting module; the case $n=0$ is the Auslander–Reiten conjecture, and the statement is related to the Wakamatsu tilting conjecture.