The BDPP conjecture for compact Kähler manifolds

Let $X$ be a compact Kähler manifold. The canonical class $K_X$ is pseudoeffective if it is represented by a closed positive $(1,1)$-current, and $X$ is uniruled if it is covered by rational curves.

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

This is the Kähler analogue of the Boucksom–Demailly–Păun–Peternell conjecture and is used in the paper to obtain a cone theorem for generalized klt pairs in arbitrary dimension. Its resolution status is not specified in the source.