Kollár's structure conjecture for varieties with big fundamental group

Let $X$ be a smooth projective variety with big fundamental group and Kodaira dimension satisfying

$$0<\kappa(X)<\dim X.$$

A fundamental group is big in the sense defined in the source. Kollár's structure conjecture. There exists a finite étale cover $p:X'\to X$ such that $X'$ is birational to a smooth family of abelian varieties over a projective variety of general type $Z$ with big fundamental group. This conjecture predicts a birational structure governed by the Iitaka fibration and the fundamental-group positivity condition; the source gives no resolution.