Kawatani's converse conjecture on minimality and derived indecomposability

Let $X$ be a smooth projective variety. A variety is derived-indecomposable if $D(X)$ has no nontrivial semi-orthogonal decompositions. Kawatani's converse conjecture. If $D(X)$ has no nontrivial semi-orthogonal decompositions, then $X$ is minimal. This is presented as the converse to the expectation that minimal varieties under suitable canonical-bundle hypotheses have indecomposable derived categories; its resolution is not indicated here.