Wu–Zheng's structure conjecture for Kähler manifolds with nef cotangent bundle

Let $X$ be a compact Kähler manifold, and suppose that its cotangent bundle $\Omega_X^1$ is nef. Let $k(X)$ denote the Kodaira dimension of $X$.

Wu–Zheng's conjecture. There exists a finite étale cover $X'$ of $X$ admitting a smooth fibration in complex tori

$$X'\to Y$$

onto a manifold $Y$ such that $\dim Y=k(X)$ and the canonical bundle $\omega_Y$ is ample.

The paper states that the preceding infinite-fundamental-group conjecture is a consequence of this conjecture in the Kähler case. Its resolution status is not specified in the supplied text.