Polishchuk–Van den Bergh conjecture on equivariant derived categories

Let $G$ be a finite group acting effectively on a smooth quasiprojective variety $X$. For $g\in G$, let $X^g$ denote the invariant subvariety and let $C(g)$ be the centralizer of $g$ in $G$. Assume that each quotient $X^g/C(g)$ is smooth. Polishchuk–Van den Bergh conjecture. There should exist a semiorthogonal decomposition

$$D^G(X)\cong\langle \mathcal{C}_{[g]}\rangle_{[g]\in \operatorname{Conj} G}$$

indexed by the conjugacy classes of $G$, with

$$\mathcal{C}_{[g]}\cong D(X^g/C(g)).$$

The cited theorem establishes this conjectural decomposition for the linear action of a real reflection group of rank $3$ on $\mathbb{A}^3$.