The nef-and-effective canonical bundle conjecture for semi-orthogonal indecomposability

Let $X$ be a smooth projective variety. A semi-orthogonal decomposition of ${\mathbf{D}}^b(X)$ is called non-trivial when it has more than one nonzero component. The canonical line bundle of $X$ is denoted by $\omega_X$.

Nef-and-effective canonical bundle conjecture. If $\omega_X$ is nef and effective, then ${\mathbf{D}}^b(X)$ has no non-trivial semi-orthogonal decompositions.

This folklore conjecture concerns the converse direction to the proposed relation between derived-category indecomposability and minimality. The source cites it as a folklore question and notes that the converse of the preceding conjecture is false, with Enriques surfaces furnishing a counterexample; the stated conjecture is marked as disproved in the supplied status information.