Log crepant derived equivalence conjecture for associated Deligne–Mumford stacks

Let $(X,B)$ and $(Y,C)$ be pairs of quasi-projective varieties with $\mathbb{Q}$-divisors satisfying the smooth-local-covering condition: there are quasi-finite surjective morphisms from smooth varieties $U$ to each variety such that the pullback of the corresponding log canonical divisor is the canonical divisor of the cover. Let $\mathcal{X}$ and $\mathcal{Y}$ be the associated Deligne–Mumford stacks. Suppose there are proper birational morphisms

$$\mu: W\to X,\qquad \nu: W\to Y$$

from a third variety $W$ such that

$$\mu^*(K_X+B)=\nu^*(K_Y+C).$$

Log crepant derived equivalence conjecture. Then there exists an equivalence of triangulated categories

$$D^b\operatorname{Coh}(\mathcal{X})\simeq D^b\operatorname{Coh}(\mathcal{Y}).$$

This is the precise common-resolution form of the logarithmic conjecture. The paper proves it for toroidal varieties and notes consequences such as the McKay correspondence for abelian quotient singularities, while the difficult general case remains open.