Stable homology conjecture for Hurwitz spaces of higher-genus curves

Let $g\geq2$ be an integer and let $G$ be a finite group. Let $\operatorname{CHur}_g^G$ be the algebraic stack over $\mathbb C$ whose objects are geometrically connected finite étale Galois $G$-covers $X\to C$ of smooth proper genus-$g$ curves, and let $\mathscr M_g$ be the moduli stack of genus-$g$ curves. For a connected component $Z\subset\operatorname{CHur}_g^G$, write $H_i(-,\mathbb Z[1/|G|])$ for homology with coefficients in $\mathbb Z[1/|G|]$. Stable homology conjecture. Fix a finite group $G$. There are constants $A$ and $B>0$, depending on $G$, such that for every connected component $Z\subset\operatorname{CHur}_g^G$ and $g>A+Bi$, the map

$$H_i(Z,\mathbb Z[1/|G|])\longrightarrow H_i(\mathscr M_g,\mathbb Z[1/|G|])$$

is an isomorphism. The conjecture predicts that the homology of every Hurwitz-space component agrees with that of $\mathscr M_g$ in a linear stability range. The case $i=0$ is tautological; the cyclic-group case is nearly established only with rational coefficients and a quadratic range, while the general conjecture remains open.