Kawamata's finiteness conjecture for Fourier–Mukai partners

Let a Fourier–Mukai partner of a smooth projective variety $X$ be a smooth projective variety $Y$ such that $\mathrm{D}^{b}(X)\cong\mathrm{D}^{b}(Y)$. Kawamata's conjecture. The number of Fourier–Mukai partners up to isomorphism of a smooth projective variety is finite. This extends the finiteness result known for projective K3 surfaces; the conjecture asks for finiteness in general.