Kawamata's splitting-functor Fourier–Mukai conjecture

Let $X_1$ and $X_2$ be smooth projective varieties. An exact functor $\mathsf{F}\colon\mathrm{D}^{b}(X_1)\to\mathrm{D}^{b}(X_2)$ is called splitting when it is both right and left splitting, where right and left splitting are defined through admissibility of its kernel and image and full faithfulness on the corresponding orthogonals. Kawamata's conjecture. Any exact splitting functor

$$\mathsf{F}\colon\mathrm{D}^{b}(X_1)\to\mathrm{D}^{b}(X_2)$$

is of Fourier–Mukai type. The source attributes this to Kawamata and records its solution in connection with other problems, so it is treated here as solved.