Minimal noncommutative desingularization conjecture for canonical singularities

Let $X$ have canonical singularities, and let $q:Y\to X$ be a finite morphism with $Y$ smooth. The pair

$$(\operatorname{id}_X,\mathcal{E}nd(q_*\mathcal{O}_Y))$$

is a noncommutative desingularization of $X$.

Minimal noncommutative desingularization conjecture. The pair $(\operatorname{id}_X,\mathcal{E}nd(q_*\mathcal{O}_Y))$ is a minimal desingularization of $X$.

The claim proposes a canonical minimal desingularization in the noncommutative interpretation of the minimal model program. The source does not provide evidence resolving it.