Vanishing higher homology conjecture for symmetric-group Hurwitz spaces

Let $G$ be a symmetric group on more than two letters, and let $c$ be the conjugacy class of transpositions. For each integer $i\geq 2$, consider the rational homology of the connected Hurwitz space $\operatorname{CHur}^c_{G,n}$. Vanishing higher homology conjecture. For every $i\geq 2$,

$$H_i(\operatorname{CHur}^c_{G,n};\mathbb{Q})$$

vanishes for sufficiently large $n$. This is proposed as a higher-homological analogue of Hurwitz's connectivity theorem and is presented as a conjectural ingredient for understanding stable homology and arithmetic asymptotics.