The $T_n$ conjecture on compact Kähler manifolds with generically large fundamental group

Let $X$ be a compact Kähler manifold of dimension $n$ with generically large fundamental group, meaning that the fundamental-group image of every subvariety through a very general point is infinite. Suppose that either

$$\text{there exists a proper modification }\widetilde{\mathbb{C}}^k\to\mathbb{C}^k\text{ and a surjective map }\widetilde{\mathbb{C}}^k\to\widetilde X,$$

or $\kappa(X)\leq 0$, where $\widetilde X$ is the universal covering. The $T_n$ conjecture. There exists a finite étale cover $X'\to X$ such that $X'$ is bimeromorphic to a torus.

This is presented as a reduction step toward Iitaka's conjecture for non-algebraic compact Kähler manifolds that are not simple. Its general validity is not established in the source.