Polishchuk–Vandenberghe conjecture on semiorthogonal decompositions of equivariant derived categories

Let $X$ be a smooth variety over an algebraically closed field of characteristic zero, and let a finite group $G$ act effectively on $X$. For $\lambda\in G/{\sim}$, let $X_\lambda$ be the invariant subvariety of $\lambda$, let $C(\lambda)$ be its centralizer, and suppose that each geometric quotient $X_\lambda/C(\lambda)$ is smooth. The pieces below are indexed by the conjugacy classes of $G$. Polishchuk–Vandenberghe conjecture. There should be a semiorthogonal decomposition of the derived category $\mathcal{D}[X/G]$ whose pieces $\mathcal{C}_{[\lambda]}$ satisfy

$$\mathcal{C}_{[\lambda]}\cong \mathcal{D}(X_\lambda/C(\lambda)).$$

This conjecture asks whether the Hochschild-homology decomposition indexed by conjugacy classes can be lifted to the level of derived categories. The source presents it as a natural question in the setting where all the geometric quotients are smooth; the supplied text gives no resolution, so its status remains open.