The Albanese fibration conjecture for compact Kähler manifolds with nef anticanonical bundle

Let $X$ be a compact Kähler manifold with nef anticanonical line bundle. Its Albanese morphism is the morphism from $X$ to its Albanese torus.

Albanese fibration conjecture. The Albanese morphism of $X$ is a locally trivial fibration with connected fibers.

The conjecture is fully proven when $X$ is projective and was proven in dimension three in the compact Kähler case. It remains widely open in higher-dimensional compact Kähler geometry, where the Albanese morphism need not be projective.