Carlson–Toledo conjecture in terms of the Hurewicz morphism

Let $\Gamma$ be an infinite Kähler group. Carlson–Toledo conjecture. There exists a compact Kähler manifold $M$ with $\pi_1(M)$ a finite-index subgroup of $\Gamma$ such that the Hurewicz morphism

$$\pi_2(M)\otimes_{\mathbb{Z}}\mathbb{R}\longrightarrow H_2(M,\mathbb{R})$$

is not surjective.

By the Hopf exact sequence, this is a topological restatement of the Carlson–Toledo conjecture on virtually positive second cohomology. The source does not state that it has been resolved.