Conjecture on transcendence of conjugacy growth series for non virtually abelian finitely presented groups

Let $G$ be a finitely presented group with a finite generating set $X$. Its conjugacy growth series is the generating function

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

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

The conjecture asserts a universal transcendence property for conjugacy growth series outside the virtually abelian case. The cited source presents it as a conjecture supported by known results, and no resolution is given here.