The DK conjecture on K-equivalent smooth varieties

Let $X_1$ and $X_2$ be smooth varieties, and let $\mu:X_1\longrightarrow X_2$ be a birational map resolved by morphisms

$$f_i:X_0\longrightarrow X_i$$

such that $f_1^*K_{X_1}$ and $f_2^*K_{X_2}$ are linearly equivalent. In this situation, $X_1$ and $X_2$ are K-equivalent. DK conjecture. K-equivalent smooth varieties should be derived equivalent:

$$D^b\operatorname{Coh}(X_1)\simeq D^b\operatorname{Coh}(X_2).$$

The conjecture, formulated independently by Bondal–Orlov and Kawamata, predicts that crepant birational transformations preserve derived categories. The paper proves derived equivalences for generalized Grassmannian flops of $D_4$ and $G_2^{\dagger}$ type, providing new evidence, while the conjecture remains open in general.