Kähler-group conjecture for mapping class groups

For $g\geq 3$, let ${\rm Mod}_g$ be the mapping class group of a closed genus-$g$ surface. Kähler-group conjecture. The group ${\rm Mod}_g$ is a Kähler group, meaning that it is isomorphic to the fundamental group of a compact Kähler manifold. The source calls this a folklore conjecture and says it was not known there; the supplied status evidence records that it is now disproved, because finite extensions reduce to pure braid groups, which are iterated extensions of free groups and are not Kähler.