Countability conjecture for horofunctions on Cayley graphs of polynomial growth

Let $G$ be a Cayley graph of polynomial volume growth. A horofunction is a function obtained as the pointwise limit associated with a geodesic ray in $G$; equivalently, for a geodesic ray $\omega=(z_1,z_2,\ldots)$, define

$$f_\omega(y)=\lim_{n\to\infty}\bigl(d(z_n,y)-d(z_n,o)\bigr),$$

where $o$ is a fixed vertex and $d$ is the graph metric.

Countability conjecture. The set of horofunctions on $G$ is countable.

This asks for a generalization of the preceding finiteness result for graphs of linear volume growth to Cayley graphs of polynomial volume growth. The source presents it as a related, probably much more difficult, question and gives no resolution.