The CAT(0) Semistability Conjecture

Let $G$ be a 1-ended CAT(0) group, meaning a group acting geometrically on a proper CAT(0) space. Its end invariant $\operatorname*{pro}$-$\pi_{1}(\varepsilon(G))$ is defined through the corresponding universal-cover or CAT(0)-space model. The CAT(0) Semistability Conjecture. Every 1-ended CAT(0) group $G$ is semistable. This is stated as an open problem because the relevant fundamental-group-at-infinity invariants are not generally known to be well-defined; many CAT(0) boundary results depend on a positive answer.