The conjugacy-ratio conjecture for finitely generated groups

Let $G$ be a group generated by a finite set $X$. Define the ball

$$\mathbb{B}_{G,X}(n)$$

of radius $n$ in the corresponding word metric, and let $C_{G,X}(n)$ be the set of conjugacy classes of $G$ having a representative in $\mathbb{B}_{G,X}(n)$. The conjugacy ratio is

$$\operatorname{CR}_X(G)=\limsup_{n\rightarrow\infty}\frac{|C_{G,X}(n)|}{|\mathbb{B}_{G,X}(n)|}.$$

Conjugacy-ratio conjecture. The group $G$ has positive conjugacy ratio if and only if $G$ is virtually abelian:

$$\operatorname{CR}_X(G)>0\quad\Longleftrightarrow\quad G\text{ is virtually abelian}.$$

This conjecture predicts that, apart from virtually abelian groups, conjugacy classes are asymptotically negligible compared with elements in every finitely generated group. The paper confirms it for certain residually finite groups of subexponential growth, hyperbolic groups, right-angled Artin groups, and the lamplighter group, but the general case remains open.