Kontsevich's $K(\pi,1)$ conjecture for strata of quadratic differentials

Let $\mathcal H$ and $\mathcal Q_g$ denote the moduli spaces of Abelian and quadratic differentials, respectively, and consider a connected component of a stratum of

$$\mathcal H\sqcup\mathcal Q_g.$$

A group $\pi$ is commensurable with a mapping class group if it has a finite-index subgroup isomorphic to a finite-index subgroup of that mapping class group. A space is a $K(\pi,1)$ if it is aspherical with fundamental group $\pi$. Kontsevich's conjecture. Each connected component of every stratum of the moduli space $\mathcal H\sqcup\mathcal Q_g$ is a $K(\pi,1)$, where $\pi$ is a group commensurable with some mapping class group. This conjecture concerns the topology of the connected components of strata; the source presents it as a conjecture, while the supplied status is unknown.