The Albanese submersion conjecture for nef anticanonical manifolds

Let $X$ be a projective manifold. The anticanonical bundle is nef when $-K_X$ has nonnegative degree on every curve. The Albanese map is the canonical morphism from $X$ to its Albanese variety. Albanese submersion conjecture. If $-K_X$ is nef, then the Albanese map is a surjective submersion. Surjectivity is known by Qi Zhang, while smoothness of the Albanese map is known in dimensions at most $3$; the assertion is open in general.