Stable Chow groups 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 introduced above. Fix an integer $i$. Suppose all $n_1,\ldots,n_k$ are sufficiently large, with the required bound depending on $i$.

Stable Chow groups conjecture. The $i$th rational Chow groups of $\operatorname{CHur}^{G,c_1,\ldots,c_k}_{n_1,\ldots,n_k}$ stabilize, in the sense that they are independent of $n_1,\ldots,n_k$. The stronger statement in the source asks whether, for every connected component $Z$ of this Hurwitz space, the branch-locus map

$$Z\longrightarrow\conf_{n_1,\ldots,n_k}$$

induces an isomorphism on the $i$th rational Chow groups. This is a more ambitious analogue of the stable homology conjecture, and the source presents it as an open question.