Refined indecomposability conjecture for varieties with positive irregularity

Let $X$ be a smooth projective variety, let $K_X$ be nef and effective, and define the irregularity by $q(X)=h^1(\mathcal{O}_X)$. Refined indecomposability conjecture. If $q(X)\geq 1$, then $D(X)$ admits no nontrivial semi-orthogonal decompositions. This refines the preceding conjecture by imposing positive irregularity, equivalently requiring $\dim \operatorname{Pic}^0(X)\geq 1$; its resolution is not indicated here.