Given $n ≥ 0$, the conjugacy growth function $c(n)$ of a finitely generated group $G$, with respect to some finite generating set $S$, counts the number of conjugacy classes intersecting the ball of radius $n$ in the Cayley graph of $G$ with respect to $S$.

Suppose $G$ is a polycyclic group that is not virtually nilpotent, then $G$ has an exponential conjugacy growth function (https://arxiv.org/pdf/1006.1064.pdf).

I was wondering if anybody has computed an upper bound of the conjugacy exponential rate (for a given generating set)?