Connectivity criterion for homological stability in homogeneous categories

Let $(\mathcal{C},\oplus,0)$ be a homogeneous category and let $A$ and $X$ be objects of $\mathcal{C}$. Set $G_n=\operatorname{Aut}(A\oplus X^{\oplus n})$, and let $W_n(A,X)_\bullet$ be the associated semi-simplicial sets. Connectivity conjecture. If the groups $G_n$ satisfy homological stability for all twisted coefficient systems as in Theorem~, then the connectivity of $W_n(A,X)_\bullet$ tends to infinity with $n$. The conjecture proposes a converse to the usual use of connectivity to prove homological stability; its status is not resolved in the supplied text.