Lower bounds for conjugacy depth functions of wreath products

Let $A$ be a finitely generated abelian group and let $G$ be a conjugacy separable group with separable cyclic subgroups. For groups with conjugacy depth function $\operatorname{Conj}$, write $f\preceq g$ when $f$ is bounded above by $g$ up to the standard comparison for conjugacy depth functions. The wreath-product lower-bound conjecture. If $A$ is finite, then

$$2^n \preceq \operatorname{Conj}_{A \wr G}(n).$$

Otherwise,

$$(\log(n))^{n} \preceq \operatorname{Conj}_{A \wr G}(n).$$

These bounds are conjectured because the examples constructed using the paper's main theorems suggest that the obtained lower bounds cannot be relaxed. The source does not provide a resolution, so the conjecture remains open.