Iitaka's uniformisation conjecture for compact Kähler manifolds

Let $X$ be a compact Kähler manifold of dimension $n$, and let $\tilde X$ be its universal covering. Assume that

$$\tilde X\simeq \mathbb{C}^n.$$

Iitaka's conjecture. Then $X$ is a torus, up to finite étale cover.

Nakayama proved this in several cases, including projective manifolds of dimension at most three; the paper establishes it in various cases in dimension four, while the general statement remains open.