Conjecture on transcendence of conjugacy growth series

Let $G$ be a finitely presented group that is not virtually abelian, and let its conjugacy growth series be the formal power series whose coefficients are the values of the conjugacy growth function. Conjugacy growth-series transcendence conjecture. The conjugacy growth series of $G$ is transcendental. The source places this conjecture alongside known rationality for virtually abelian groups and transcendence results for several broader classes; the general finitely presented case remains open in the source.