Kuznetsov's Fourier–Mukai conjecture for categorical data

Let $X$ and $Y$ be irreducible smooth projective schemes over $\mathbb{C}$, of dimensions $d_X$ and $d_Y$, respectively, related by the categorical data in the source. Let ${\mathcal T}_X$ and ${\mathcal T}_Y$ be the associated subcategories, let $\phi:{\mathcal T}_X\simeq {\mathcal T}_Y$ be an equivalence, and let

$$\Phi:\operatorname{perf}(X)\to\operatorname{perf}(Y)$$

be the composition of the projection onto ${\mathcal T}_X$, the equivalence $\phi$, and the inclusion into $\operatorname{perf}(Y)$. Kuznetsov's Fourier–Mukai conjecture. There exists a perfect complex ${\mathcal E}\in\operatorname{perf}(X\times Y)$ such that $\Phi$ is isomorphic to

$$\Phi_{\mathcal E}(-):=Rq_*(p^*(-)\otimes^L{\mathcal E}),$$

where $p:X\times Y\to X$ and $q:X\times Y\to Y$ are the projection morphisms. This is a representability assertion for the functors arising from the categorical data; the supplied text gives no resolution status.