Carlson–Toledo conjecture on virtually positive second Betti number of Kähler groups

Let $G$ be an infinite Kähler group, meaning the fundamental group of a smooth complex Kähler manifold. Carlson–Toledo conjecture. There exists a finite-index subgroup $G^\circ$ of $G$ such that its second Betti number $b_2(G^\circ)$ is positive. This conjecture predicts a virtual cohomological property for infinite Kähler groups; the source presents it as a conjecture and gives no resolution status.