Kuznetsov's conjecture on minimal Lefschetz decompositions and categorical resolutions

Let $X$ be a smooth projective variety. A minimal Lefschetz decomposition of $\D^b(X)$ is a Lefschetz decomposition whose first block is minimal under inclusion, and a categorical resolution of singularities is a categorical analogue of a resolution of the affine cone over $X$. Kuznetsov's conjecture. Minimal Lefschetz decompositions of $\D^b(X)$ correspond to minimal categorical resolutions of singularities of the affine cone over $X$. This relates minimal semi-orthogonal decompositions of derived categories to categorical resolutions of the cones over projective varieties; the source gives no resolution of the conjecture.