Ciobanu–Evetts conjecture on conjugacy growth series

Let $G$ be a finitely presented group, and let $X$ be a finite generating set. The conjugacy growth series of $G$ with respect to $X$ is the generating function

$$\sum_{n\geq 0} c_{G,X}(n) t^n,$$

where $c_{G,X}(n)$ counts the conjugacy classes having a minimal-length representative of length $n$ with respect to $X$. Ciobanu–Evetts conjecture. If $G$ is not virtually abelian, then its conjugacy growth series is transcendental.

Virtually abelian groups are known to have rational conjugacy growth series with respect to all generating sets, while the conjecture predicts transcendence for every finitely presented group that is not virtually abelian.