The projective h-principle fundamental-group conjecture

Let $X$ be a complex projective manifold satisfying the $h$-principle, and let $\pi_1(X)$ be its fundamental group. A group is almost abelian if it contains an abelian subgroup of finite index.

Projective h-principle fundamental-group conjecture. If $X$ is a complex projective manifold satisfying the $h$-principle, then $\pi_1(X)$ is almost abelian.

This is stated as a consequence of the theorem proving that such a projective manifold is special together with the conjectured almost abelianity of the fundamental group of a special manifold. It remains conditional and unresolved in the source.