Kawamata's categorical resolution conjecture for K-equivalent stacks

Let $\mathcal{X}$ and $\mathcal{Y}$ be smooth Deligne--Mumford stacks with Gorenstein moduli spaces $X$ and $Y$. Suppose there are birational maps

$$f:Z\to X,\qquad g:Z\to Y,$$

such that $f^*K_X=g^*K_Y$. The Categorical Resolution Conjecture. The derived categories of sheaves in the étale topology are equivalent:

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

This is a stack-theoretic form of the expectation that crepant or K-equivalent birational models have equivalent derived categories. The source attributes it to a conjecture of Kawamata and presents the displayed formulation as a slightly generalized part of his Conjecture 1.2; no resolution status is given here.