3 problems

Generic horofunction boundary conjecture. If $S$ is empty, the horofunction boundary is as in the stated theorem: it has exactly two Busemann points and countably many non-Busemann…

Subexponential-growth finite-orbit conjecture. The horofunction boundary of $\Gamma$ contains a finite orbit for the underlying group action.

Finite-orbit conjecture. For any Cayley graph of $G$, the horofunction boundary contains a finite orbit for the canonical group action.