The derived-category conjecture for four-dimensional $X_{10}$ varieties

Let $X$ be a smooth four-dimensional variety of type $X_{10}$. Let $D^b(X)$ denote its bounded derived category, and let $K3$ denote a generic noncommutative K3 surface. The objects $E_1,E_2,E_3,E_4$ are exceptional objects.

Derived-category conjecture for $X_{10}$.

$$D^b(X)=\langle D^b(K3),E_1,E_2,E_3,E_4\rangle.$$

The conjecture is expected to follow from a version of homological projective duality and is presented as an analogue of the known decomposition for cubic fourfolds.