Subexponential-growth horofunction-boundary orbit conjecture

Let $\Gamma$ be a Cayley graph, and let the size of its ball of radius $r$ be bounded by

$$C \exp(Cr^\alpha)$$

for some $C>0$, some $\alpha<\tfrac12$, and every $r\in\mathbb{N}$. The horofunction boundary of $\Gamma$ carries the underlying group's natural action.

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

This is presented as a logically weaker conjecture than the polynomial-growth finite-orbit conjecture and is motivated by the connection between Busemann points, geodesics, and growth. The source does not establish it.