Indecomposability conjecture for varieties with nef effective canonical bundle

Let $X$ be a smooth projective variety. A canonical bundle is nef and effective when $K_X$ is nef and has a nonzero global section. A semiorthogonal decomposition of $D^b_{\!\operatorname{coh}}(X)$ is non-trivial when it has more than one nonzero component.

Indecomposability conjecture. If the canonical bundle $K_X$ is nef and effective, then $X$ admits no non-trivial semiorthogonal decompositions.

This conjecture predicts indecomposability of derived categories for varieties whose canonical bundle has both numerical positivity and a section. The paper proves the claim under the stronger assumption that the base locus of $|K_X|$ has dimension at most one, while the general case remains open.