Pseudoeffectivity and uniruledness conjecture for compact Kähler manifolds

Let $X$ be a compact Kähler manifold. A compact complex manifold is uniruled if it is covered by rational curves.

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

This conjecture extends the corresponding relationship between pseudoeffectivity of the canonical class and uniruledness beyond the projective setting. The paper presents an inductive strategy toward the existence of rational curves on compact Kähler manifolds, but the conjecture remains open in the stated generality.