Revised Campana conjecture on virtually nilpotent fundamental groups

Let $X$ be a smooth quasi-projective variety. It is special or $h$-special according to the notions defined in the source, and a group is virtually nilpotent if it has a nilpotent subgroup of finite index. Revised Campana conjecture. If $X$ is special or $h$-special, then its fundamental group is virtually nilpotent. The revision is motivated by counterexamples to virtual abelianity; the source gives no resolution of this revised claim.