Finite horofunction-boundary orbit conjecture for polynomial-growth groups

Let $G$ be a finitely generated group of polynomial growth. Its Cayley graphs have horofunction boundaries, with the canonical group action induced by left multiplication.

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

This would extend the use of horofunctions to detect virtual characters beyond the established polynomial-growth results. The statement is also related to Conjecture 1.3 in the cited work, but its resolution is not given here.