The pseudoeffectivity criterion for uniruled compact Kähler manifolds

Let $X$ be a compact Kähler manifold. The canonical class $K_X$ is pseudoeffective if its numerical class lies in the pseudoeffective cone, and $X$ is uniruled if it is covered by rational curves.

Pseudoeffectivity–uniruledness conjecture. The canonical class $K_X$ is pseudoeffective if and only if $X$ is not uniruled.

This conjecture would reduce the existence of rational curves on higher-dimensional compact Kähler manifolds to a characterization of uniruledness. The supplied text gives no resolution status.