Some background (see e.g. the books by Ghys & de la Harpe or Bridson & Haefliger for more information):
Let $\Gamma$ be a group with a finite symmetric generating set $S$. Recall that $\Gamma$ is called a (word) hyperbolic group if the Cayley graph of $(\Gamma, S)$ is hyperbolic (in the sense of Gromov). This notion is independent of the choice of the generating set $S$.
Assuming $\Gamma$ to be hyperbolic, there is a finite directed graph canonically associated to $(\Gamma,S)$ called the geodesic automaton (GA), whose arrows are labelled by elements of $S$. Paths in the GA graph starting from a special basepoint correspond to segments of geodesics in the Cayley graph.
Now consider the recurrent vertices of the GA graph, i.e., the set of vertices that belong to loops. Following this paper by Haissinsky, Mathieu, and Mueller, let us say that the GA graph is strongly connected if given any two recurrent vertices, there is one path from one to the other.
Question. If $\Gamma$ is a nonelementary hyperbolic group, is the geodesic automaton graph strongly connected for some (all?) choice of $S$?