Sphere-boundary graphical discreteness conjecture in dimensions at least four

Let $G$ be a hyperbolic group, and let $\partial G$ denote its boundary. Assume that $\partial G$ is homeomorphic to the $n$-sphere $S^n$, where $n\geq 4$. A group is graphically discrete when it has only trivial lattice envelopes.

Sphere-boundary graphical discreteness conjecture. Hyperbolic groups with boundary the $n$-sphere for $n\geq 4$ are graphically discrete.

The paper proves graphical discreteness for sphere boundaries when $n\leq 3$ and conjectures the corresponding statement for $n\geq 4$.