Kuznetsov's categorical McKay conjecture for quotient singularities

Let $V$ be a smooth quasi-projective variety, and let $G$ be a finite subgroup of $\operatorname{Aut}(V)$ such that $\omega_V$ is $G$-equivariantly locally trivial. Let $Z\to V/G$ be any resolution of singularities. Kuznetsov's conjecture. There exists an admissible fully faithful embedding

$$\mathrm{D}^b(\operatorname{Coh}^G(V))\hookrightarrow \mathrm{D}^b(Z).$$

In particular, if $Z$ is a crepant resolution of $V/G$, there is an equivalence

$$\mathrm{D}^b(\operatorname{Coh}^G(V))\simeq\mathrm{D}^b(Z).$$

This is the stated categorical form of the McKay correspondence for quotient singularities. The last equivalence is known when $\dim V\leq 3$, or when $V$ is symplectic and $G$ acts by symplectic automorphisms; the general assertion remains open.