The nef anticanonical Albanese submersion conjecture

Let $X$ be a projective manifold with nef anticanonical divisor $-K_X$.

Albanese submersion conjecture. The Albanese map of $X$ is a surjective submersion.

The surjectivity is known in the projective case by work of Qi Zhang, while the statement is known in dimension $3$ and when $-K_X$ is Hermitian semipositive or admits a singular metric with trivial multiplier ideal. The corresponding surjectivity statement remains unknown in the Kähler case.