Kawamata's stacky DK conjecture for varieties with finite quotient singularities

Let $X$ and $Y$ be projective varieties with only finite quotient singularities, and let $\mathcal{X}(X)$ and $\mathcal{X}(Y)$ be their associated stacks. The varieties $X$ and $Y$ are $K$-equivalent when their canonical divisors agree after pullback to a common resolution. Kawamata's stacky DK conjecture. The varieties $X$ and $Y$ are $K$-equivalent if and only if there is an equivalence of triangulated categories

$$D^{\mathrm{b}}\bigl(\operatorname{Coh}(\mathcal{X}(X))\bigr)\cong D^{\mathrm{b}}\bigl(\operatorname{Coh}(\mathcal{X}(Y))\bigr).$$

This is a stack-theoretic extension of the Bondal–Orlov–Kawamata DK conjecture, relating birational canonical data to derived categories. The source presents it as a conjectural formulation for projective varieties with finite quotient singularities; no resolution is given here.