Uniform Fourier–Mukai kernel bound conjecture equivalent to Kawamata's finiteness conjecture

Let $(X_i)_{i\in\mathbb N}$ be the set of varieties derived equivalent to a fixed smooth projective variety $X$. For each $i$, let $\mathcal A_i$ be an ample line bundle on $X_i$, let $F_i:D^b(X)\to D^b(X_i)$ be a triangulated equivalence, and let $\mathcal K_i^\cdot\in D^b(X\times X)$ be the kernel of the Fourier–Mukai transform $F_i^*(\mathord{-}\otimes\mathcal A_i)$. Let $\mathcal L$ be a very ample line bundle on $X\times X$, and choose $d\in\mathbb N$ such that $\mathcal L\oplus\mathcal L^{-1}\oplus\cdots\oplus\mathcal L^{-d}$ is a strong generator of $D^b_{qc}(X\times X)$. Write $\mathbb H^j$ for hypercohomology.

Uniform Fourier–Mukai kernel bound conjecture. The line bundles $\mathcal A_i$ and equivalences $F_i$ can be chosen so that there exists a finitely supported function $\nu:\mathbb Z\to\mathbb N$ satisfying

$$\dim\mathbb H^j(X,\mathcal K_i^\cdot\otimes\mathcal L^l)\leq\nu(j)\quad\forall j,\ \forall l\in[0,d],\ \forall i.$$

The source states that this condition is equivalent to the preceding finiteness conjecture, rather than presenting it as an independent claim. Its resolution status is not specified in the source.