The KD hypothesis for quotient-type pairs

Let $(X,B)$ and $(Y,C)$ be pairs of quotient type with $X$ and $Y$ projective. Let $\widetilde X$ and $\widetilde Y$ be their associated smooth Deligne–Mumford stacks. Define $K$-equivalence and the $K$-inequality by requiring, on a common projective birational model $Z$, respectively $f^*(K_X+B)=g^*(K_Y+C)$ and $f^*(K_X+B)\leq g^*(K_Y+C)$.

KD hypothesis. If $(X,B)\sim_K(Y,C)$, then there is an equivalence

$$\Phi:D^b\operatorname{Coh}(\widetilde X)\cong D^b\operatorname{Coh}(\widetilde Y).$$

If $(X,B)\leq_K(Y,C)$, then there is a fully faithful functor

$$\Phi:D^b\operatorname{Coh}(\widetilde X)\to D^b\operatorname{Coh}(\widetilde Y).$$

This extends the derived-category predictions from smooth varieties to quotient-type KLT pairs using associated smooth Deligne–Mumford stacks. The source presents it as a hypothesis and gives no resolution.