The minimal categorical resolution conjecture

Let $Y$ be a quasiprojective scheme. A categorical resolution of $Y$ is a smooth triangulated category $\mathcal{T}$ equipped with adjoint functors $\pi_*:\mathcal{T}\to D^b\operatorname{Coh}(Y)$ and $\pi^*:D^\mathrm{perf}(Y)\to\mathcal{T}$ such that $\pi_*\circ\pi^*$ is isomorphic to the identity on $D^\mathrm{perf}(Y)$. Say that one categorical resolution dominates another if there is a fully faithful functor between their categories compatible with the pushforward functors. Minimal categorical resolution conjecture. For any quasiprojective scheme $Y$ there exists a categorical resolution which is minimal with respect to the dominance order. This is proposed as a categorical analogue of minimal-model constructions; the source gives no resolution.