The structure conjecture for compact Kähler manifolds with nef anti-canonical bundle

Let $X$ be a compact Kähler manifold with nef anti-canonical bundle $-K_X$. A locally constant fibration is a fibration that is locally constant in the sense of Definition 2.3 of Matsumura–Wang. The structure conjecture. There exists a fibration $f\colon X\to Y$ such that: (1) $f\colon X\to Y$ is a locally constant fibration; (2) the fiber $F$ is a rationally connected manifold; and (3) the base $Y$ is a compact Kähler manifold with $c_1(Y)=0$. This conjecture was resolved by Cao–Höring for smooth projective varieties and generalized to projective klt pairs by Matsumura–Wang; in the setting stated here, the source identifies it as resolved.