The Abelianity Conjecture for special compact Kähler manifolds

A special compact Kähler manifold is a compact Kähler manifold belonging to the class of special manifolds in the core-fibration decomposition. Its fundamental group is denoted by $\pi_1(X)$.

Abelianity Conjecture. Every special compact Kähler manifold $X$ has an almost abelian fundamental group; that is, $\pi_1(X)$ has an abelian subgroup of finite index.

The conjecture is part of the expected common structure of special manifolds, rationally connected manifolds, and manifolds with Kodaira dimension zero. The supplied status evidence indicates that this conjecture has been resolved, although the excerpt does not specify the resolving result.