Campana's almost abelian fundamental group conjecture for special varieties

Let $X$ be a special variety. Write $\pi_1(X)$ for its fundamental group, and define the maximal irregularity by

$$\widetilde q(X)=\operatorname{sup}\left\{q(\widetilde X)\mid \widetilde X\to X\text{ is finite étale}\right\}.$$

Here a group is almost abelian if it contains an abelian subgroup of finite index. Campana's fundamental-group conjecture. One has

1. $\pi_1(X)$ is almost abelian;
2. if $\widetilde q(X)=0$, then $\pi_1(X)$ is finite.

The conjecture is motivated by the expectation that special varieties are, in a suitable birational sense, successive extensions of orbifolds with Fano or numerically trivial canonical bundle. The source does not specify a resolution of either assertion.