The $L_2$-intersection cohomology concentration conjecture for analytic subsets of Kähler hyperbolic manifolds

Let $Z$ be a singular compact complex space, let $\pi:\widetilde{Z}\to Z$ be its universal covering space, and write $\mathcal{IC}^{\bullet}_Z$ for the intersection cohomology sheaf of $Z$. For a compact Kähler hyperbolic manifold and a closed analytic subset $Z$ with universal cover $\widetilde Z$, consider its $L_p$-intersection cohomology $\mathbb{H}^k_{(p)}(\widetilde{Z},\pi^{-1}\mathcal{IC}^{\bullet}_Z)$. The $L_2$-intersection cohomology concentration conjecture. If $p=2$, then

$$\mathbb{H}^k_{(2)}(\widetilde{Z},\pi^{-1}\mathcal{IC}^{\bullet}_Z)=0$$

for $k\not=\dim(Z)$. This would extend the expected concentration of $L_2$-intersection cohomology to closed analytic subsets of compact Kähler hyperbolic manifolds; the source presents it as the initial impetus for the work and gives no resolution.