Claudon–Höring–Kollár conjecture on projective varieties with quasi-projective universal cover

Let $X$ be a complex projective manifold with infinite fundamental group, and let $\widetilde{X}$ be its universal cover. Suppose that $\widetilde{X}$ is quasi-projective. Claudon–Höring–Kollár conjecture. After replacing $X$ by a finite étale cover, there exists a locally trivial fibration $X\to A$ onto a complex torus $A$ with simply connected fiber $F$. In particular,

$$\widetilde{X}\simeq F\times\mathbb{C}^{\operatorname{dim} A}.$$

This is a structural conjecture for projective manifolds with quasi-projective universal covers; the source gives no resolution.