The locally constant rationally connected fibration conjecture for Kähler manifolds with nef anticanonical bundle

Let $X$ be a compact Kähler manifold with nef anticanonical bundle. A locally constant fibration is a fibration

$$\varphi:X \simeq (\widetilde{Y} \times F)/\pi_1(Y) \to Y=\widetilde{Y}/\pi_1(Y)$$

induced by the natural projection, where $\widetilde{Y}$ is the universal cover of $Y$, $F$ is the fiber, and the fibration is equipped with a representation $\rho:\pi_1(X)\to\operatorname{Aut}(F)$.

Structural conjecture. There exists a fibration $\varphi:X\to Y$ such that $\varphi$ is locally constant, $Y$ is a compact Kähler manifold with $c_1(Y)=0$, and $F$ is rationally connected.

This structural conjecture is proven up to dimension three and is settled completely in the projective case. It predicts a locally constant rationally connected fibration over a compact Kähler base with trivial first Chern class.