Analytic continuation conjecture for the topo-symmetric extension zeta function

Let $G$ be a group and $H$ an abelian group. Let $\mathrm{Ext}_{\mathrm{ts}}(G,H)$ denote the set of topo-symmetric extensions, and let $\mathrm{ord}_{\max}(E)$ be the maximal order of elements in $E$. Define the generating function

$$\zeta_{\mathrm{ts}}(s;G,H):= \sum_{E \in \mathrm{Ext}_{\mathrm{ts}}(G,H)} \mathrm{ord}_{\max}(E)^{-s}.$$

Analytic continuation conjecture. The function $\zeta_{\mathrm{ts}}(s;G,H)$ admits an analytic continuation to the complex plane, with poles reflecting the arithmetic structure of $G$ and $H$. This proposes an analytic framework for studying topo-symmetric extensions, but no proof or precise description of the poles is supplied.