The torus quotient conjecture for Stein canonical extensions

Let $M$ be a compact Kähler manifold that is not uniruled, and let $Z_M$ be a canonical extension associated with a Kähler class on $M$. Torus quotient conjecture. If $Z_M$ is Stein for some Kähler class, then $M$ is an étale quotient of a torus. The result is known for projective manifolds of dimension at most three, where the conclusion is strengthened to an étale quotient of an abelian variety. The conjecture remains open in the stated generality.