Matsumura–Wu 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 fibration is a surjective holomorphic map with connected fibers, and it is locally constant when its fibers are locally biholomorphic in the appropriate local trivializations. A compact Kähler manifold $Y$ has $c_1(Y)=0$ when its first Chern class vanishes, and a variety is rationally connected when any two general points can be joined by a rational curve. Matsumura–Wu's conjecture. There exists a fibration

$$\varphi\colon X\to Y$$

that is locally constant, where $Y$ is a compact Kähler manifold with $c_1(Y)=0$ and the fiber $F$ of $\varphi$ is rationally connected.

This conjecture extends structure results for manifolds with nef tangent bundles and for manifolds with non-negative holomorphic bisectional curvature. The supplied text does not indicate whether it has been resolved.