Demailly–Peternell–Schneider conjecture on Albanese maps with nef anticanonical bundle

Let $X$ be a compact Kähler manifold such that $-K_X$ is nef, and let $p:X\rightarrow Y$ be its Albanese map. Demailly–Peternell–Schneider conjecture. The map $p$ is locally trivial: for any small open set $U\subset Y$, $p^{-1}(U)$ is biholomorphic to the product $U\times F$, where $F$ is the generic fibre of $p$. In particular, $p$ is a submersion. This conjecture was proved for projective manifolds in the paper itself, and was already known under stronger assumptions such as nefness of $T_X$, hermitian positivity of $-K_X$, or bigness of the anticanonical bundle of the generic fibre; the general compact Kähler case is therefore the remaining broader setting.