The $CZ_n$ conjecture on torus submersions

Let $X$ be a compact Kähler manifold of dimension $n$ with generically large fundamental group.

The $CZ_n$ conjecture. Up to finite étale cover, $X$ is bimeromorphic to a torus submersion over a variety of general type.

This more general conjecture is attributed in the source to Campana and Zhang. The source does not state a resolution, so the conjecture remains open here.