Compactifiable universal cover conjecture for compact Kähler manifolds

Let $X$ be a compact Kähler manifold with infinite fundamental group $\pi_1(X)$. Suppose that its universal cover $\widetilde{X}_{\operatorname{univ}}$ is a Zariski open subset

$$\widetilde{X}_{\operatorname{univ}}\subset\overline{X}$$

of some compact complex manifold $\overline{X}$. A finite étale cover of $X$ is understood in the assertion below.

Compactifiable universal cover conjecture. After passing to a finite étale cover, there exists a locally trivial fibration $X\rightarrow A$ with simply connected fibre $F$ onto a complex torus $A$. In particular,

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

This conjecture generalises Iitaka's classical conjecture that a compact Kähler manifold uniformised by $\mathbb{C}^{\dim X}$ is an étale quotient of a complex torus; it predicts, in particular, strong restrictions on the fundamental group of a compact Kähler manifold with compactifiable universal cover.