The hyperbolic k-geodeticity conjecture

Let $G$ be a hyperbolic group. A group is geodetic if it admits a generating set whose Cayley graph is geodetic, and $G$ is $k$-geodetic if it is $k$-geodetic for a positive integer $k$. For a finite generating set $S$, write $\operatorname{Cay}(G,S)$ for the Cayley graph of $G$ with respect to $S$.

Hyperbolic k-geodeticity conjecture. A hyperbolic group $G$ is $k$-geodetic for some positive integer $k$ if and only if $G$ is geodetic. Furthermore, if $G$ is infinite and $\operatorname{Cay}(G,S)$ is $k$-geodetic for some finite generating set $S$ and some $k\geq 1$, then $\operatorname{Cay}(G,S)$ is geodetic.

This conjecture asks whether, among hyperbolic groups, the apparently weaker bounded geodetic condition is equivalent to geodeticity, and whether the conclusion holds for every finite generating set of an infinite group. The preceding results establish structural restrictions on hyperbolic $k$-geodetic groups, including virtual freeness and cyclic centralisers of infinite-order elements, but do not resolve the conjecture.