  We construct a Cayley graph $\mathbf{Cay}_S(\Gamma)$ of a hyperbolic group
$\Gamma$ such that there are elements $g,h\in\Gamma$ and a point $\gamma \in
\partial_\infty\Gamma = \partial_\infty\mathbf{Cay}_S(\Gamma)$ such that the
sets $\mathcal{RB}(g,\gamma)$ and $\mathcal{RB}(h,\gamma)$ in
$\mathbf{Cay}_S(\Gamma)$ of vertices along geodesic rays from $g,h$ to $\gamma$
have infinite symmetric difference; thus answering a question of Huang, Sabok
and Shinko.
