The derived-category conjecture for Kuechle fourfolds

Let $X$ be a smooth four-dimensional Kuechle manifold. Let $D^b(X)$ denote its bounded derived category, let $K3$ denote a generic noncommutative K3 surface, and let $E_1,\ldots,E_n$ be the exceptional objects appearing in the proposed decomposition.

Derived-category conjecture for Kuechle fourfolds.

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

The conjecture is presented as a consequence of a possible generalization of Kuznetsov's homological projective duality; the source does not specify $n$.