Existence conjecture for minimal Lefschetz decompositions

Let $X$ be a smooth projective variety and let $L$ be an ample line bundle on $X$. A Lefschetz decomposition of $\mathit{D}^b(X)$ is minimal when it is strict and minimal with respect to the inclusion partial ordering of strict Lefschetz decompositions. Existence conjecture. There exists a minimal Lefschetz decomposition of $\mathit{D}^b(X)$ for every smooth projective variety $X$ and every ample line bundle $L$. This would guarantee the existence of a minimal Lefschetz decomposition in the stated geometric setting; the source gives no resolution of the assertion.