Cohomological Carlson–Toledo conjecture for finite étale covers

Let $M$ be a compact Kähler manifold with infinite fundamental group $\Gamma$, and let $\widetilde{M}$ denote its universal cover. Consider the direct limit over the projective system of finite étale covers $M'$ of $M$:

$$\varinjlim_{M'} H^2(M',\mathbb{R})\longrightarrow H^2(\widetilde{M},\mathbb{R}).$$

Cohomological Carlson–Toledo conjecture. This natural map is not injective.

The source presents this as the cohomological version of the stronger Hurewicz-form conjecture. Its resolution status is not given.