Stable homology conjecture for Hurwitz spaces

Let $G$) be a finite group, let $c=c_1\cup\cdots\cup c_k\subset G$ be a union of conjugacy classes, and let $\operatorname{CHur}^{G,c_1,\ldots,c_k}_{n_1,\ldots,n_k}$ be the Hurwitz space of connected $G$-covers of $\mathbb A^1_{\mathbb C}$ with a trivialization at infinity and $n_i$ branch points having inertia in $c_i$. Let $\conf_{n_1,\ldots,n_k}$ be the corresponding multicolored configuration space. Fix an integer $i$. For all $n_1,\ldots,n_k$ sufficiently large, with the required bound depending on $i$, and for every connected component $Z\subset\operatorname{CHur}^{G,c_1,\ldots,c_k}_{n_1,\ldots,n_k}$, the branch-locus map induces an isomorphism

$$H_i(Z;\mathbb Q)\xrightarrow{\sim}H_i(\conf_{n_1,\ldots,n_k};\mathbb Q).$$

Stable homology conjecture. The only stable homology of these Hurwitz spaces should be the obvious stable homology coming from the configuration space. The conjecture is known for abelian $G$ and for the nonabelian cases proved in the paper, but in general it is not even known whether the homology stabilizes, except when $k=1$ and $c_1$ is non-splitting. It is related to conjectures on counting $G$-extensions of number fields.