Strong Carlson–Toledo conjecture for finite étale covers

Let $M$ be a compact Kähler manifold with infinite fundamental group $\Gamma$. Strong Carlson–Toledo conjecture. There exists a finite étale cover $M'$ of $M$ such that the Hurewicz morphism

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

is not surjective.

This is stated as a stronger form of the Carlson–Toledo conjecture, replacing the existence of a suitable Kähler manifold by a finite étale cover of a given one. The source gives no resolution status.