Beauville–Bogomolov decomposition conjecture for manifolds with nef anticanonical class

Let $X$ be a compact Kähler manifold with nef anticanonical class. Denote its universal cover by $\widetilde{X}$. Decomposition conjecture. The universal cover decomposes as a product

$$\widetilde{X} \simeq \mathbb{C}^q \times \prod Y_j \times \prod S_k \times Z,$$

where the $Y_j$ are irreducible Calabi–Yau manifolds, the $S_k$ are irreducible hyperkähler manifolds, and $Z$ is a rationally connected manifold. This is an analogue of the Beauville–Bogomolov decomposition theorem. The conjecture has been proved under the stronger assumptions that $T_X$ is nef, $-K_X$ is hermitian semipositive, or the general fibre of the Albanese map is weak Fano.