Campana's torus-submersion conjecture for Stein universal covers

Let $X_n$ be a projective manifold whose universal cover $\tilde X$ is Stein, and assume

$$K_X^n=0.$$

Campana's torus-submersion conjecture. Up to a finite étale cover of $X$, the manifold $X$ admits a torus submersion over a projective manifold $Y$ such that $K_Y$ is ample and the universal cover of $Y$ is again Stein. The conjecture proposes a structural description of the remaining case in the preceding canonical-divisor criterion, where the canonical class is nef but not big. The supplied text gives no resolution status.