Demailly–Peternell–Schneider conjecture on the Albanese morphism of compact Kähler manifolds

Let $X$ be a compact Kähler manifold. Assume that $-K_X$ is nef.

Demailly–Peternell–Schneider conjecture. The Albanese morphism of $X$ is surjective.

This conjecture concerns the relationship between nef anticanonical bundles and the global irregularity of compact Kähler manifolds. The supplied source does not state whether it has been resolved.