Kawamata's DK hypothesis for K-equivalent varieties

Let $X$ and $Y$ be smooth projective varieties. They are $K$-equivalent if there is a crepant birational map between them, and they are Fourier–Mukai partners if their perfect derived categories are equivalent by a Fourier–Mukai transform. Kawamata's DK hypothesis. If $X$ and $Y$ are $K$-equivalent, then they are Fourier–Mukai partners; consequently, $Y$ is an open subscheme of $\operatorname{Spec}^{\mathsf{FM}}\operatorname{Perf}X$ and conversely. Conversely, if $X$ and $Y$ are birationally equivalent Fourier–Mukai partners with non-negative Kodaira dimension, then they are $K$-equivalent. The source presents this as a central open conjecture and explains that its first direction would follow from factorization into flops together with the Orlov–Bondal flop conjecture.