The structure conjecture for compact Kähler manifolds with semi-positive holomorphic sectional curvature

Let $X$ be a compact Kähler manifold with semi-positive holomorphic sectional curvature. Structure conjecture. There should exist a smooth locally trivial morphism $X\to Y$ whose fibers $F$ are projective and rationally connected, with $Y$ a compact Kähler manifold admitting a flat metric. Moreover, there should exist a complex torus $T$ and a finite étale cover $T\to Y$ such that $X^*=X\times_YT\to T$ is locally trivial with rationally connected fiber $F$. If $X_{\rm univ}$ denotes the universal cover of $X$, one should also have $X_{\rm univ}\cong\mathbb C^m\times F$, and the fundamental group of $X$ should be an extension of a finite group by $\mathbb Z^{2m}$. This is presented as a revised structure conjecture generalizing the Howard–Smyth–Wu structure theorem. The paper states that it is proved when a Mori contraction (MRC) fibration can be chosen as a morphism without indeterminacy, in particular when $X$ has nef anticanonical bundle, and for compact Kähler surfaces; the general conjecture remains open.