Rivin's rationality conjecture for spherical conjugacy growth series

Let $G$ be a word hyperbolic group. Its spherical conjugacy growth series is denoted by $\widetilde{\sigma}(G)$. Rivin's conjecture. The series $\widetilde{\sigma}(G)$ is rational if and only if $G$ is virtually cyclic.

Rivin proved that the spherical conjugacy growth series is not rational for nonabelian free groups with respect to their free generating sets. The conjecture asserts that virtual cyclicity is precisely the condition guaranteeing rationality for word hyperbolic groups; one direction is proved in the source, while the converse remains open here.