The abundance conjecture for compact Kähler manifolds

Let $X$ be a compact Kähler manifold with $K_X$ nef. A line bundle is semiample if some positive tensor power is globally generated.

Abundance conjecture. $K_X$ is semiample.

The conjecture predicts the converse to the fact that semiample line bundles are nef. It is open in general for compact Kähler manifolds, including the intermediate Kodaira-dimension case discussed here.