The virtually nilpotent fundamental-group conjecture for special and h-special varieties

Let $X$ be a special or $h$-special smooth quasi-projective variety, where $h$-special means hyperbolically special in the sense that the relation generated by chains of entire curves is Zariski dense in $X\times X$.

Virtually nilpotent fundamental-group conjecture. The fundamental group $\pi_1(X)$ is virtually nilpotent.

This is the paper's revised form of Campana's conjecture after a counterexample to virtual abelianity. The paper proves that every linear representation of $\pi_1(X)$ has virtually nilpotent image, but the full assertion about $\pi_1(X)$ itself remains open in the stated generality.