The h-principle and almost abelianity of the fundamental group

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

The h-principle fundamental-group conjecture. Every projective manifold satisfying the $h$-principle should have an almost abelian fundamental group.

This is proposed as a purely topological obstruction to the $h$-principle, combining the theorem that projective manifolds satisfying the $h$-principle are special with the cited Abelianity conjecture. Its resolution is not specified in the source.