Kawamata's fully faithful derived embedding conjecture for log K-related pairs

Let $(X,B)$ and $(Y,C)$ be pairs of quasi-projective varieties with $\mathbb{Q}$-divisors such that there exist quasi-finite and surjective morphisms

$$\pi:U\to X,\qquad \sigma:V\to Y$$

from smooth, possibly reducible, varieties satisfying

$$\pi^*(K_X+B)=K_U,\qquad \sigma^*(K_Y+C)=K_V.$$

Let $\mathcal{X}\to X$ and $\mathcal{Y}\to Y$ be the natural morphisms from 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

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

Kawamata's fully faithful embedding conjecture. Under these assumptions, there exists a fully faithful exact functor

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

This is a general conjectural form of derived embedding for log $K$-related varieties, extending the expected derived equivalence for $K$-equivalent varieties. Its status is not resolved in the supplied source.