The $H_1$-semistability conjecture for finitely presented groups

Let $G$ be a 1-ended finitely presented group, and let $\varepsilon(G)$ denote its end. Write $\operatorname*{pro}$-$H_{1}(\varepsilon(G);\mathbb{Z})$ for the inverse system of first homology groups at infinity. The $H_1$-semistability conjecture. For every 1-ended finitely presented group $G$, $\operatorname*{pro}$-$H_{1}(\varepsilon(G);\mathbb{Z})$ is semistable. The source presents this as an open variation of the Semistability Conjecture; no general resolution is given.