Conjecture on poles of abelian extension counting Dirichlet series

Let $G$ be an abelian group, and let $D_{\Q,G}(s)$ be the generating Dirichlet series for $G$-extensions of $\Q$ ordered by discriminant. Let $a(G)$ denote the minimal index, let $\ind(G-\{1\})$ be the set of indices of nonidentity elements, and let $\Phi(G)$ be the Frattini subgroup. For each relevant group $H$, define $\bar{b}_d(\Q,H)$ as in the nonvanishing theorem.

Pole-order conjecture. The series $D_{\Q,G}(s)$ should have a meromorphic continuation to

$$\operatorname{Re}(s)>\frac{1}{2a(G)}$$

with poles at $s=1/d$ for every $d\in\ind(G-\{1\})$, each of order exactly

$$\max_{\Phi(G)\le H\le G}\bar{b}_{d/[G:H]}(\Q,H).$$

This conjecture extends the established meromorphic-continuation and nonvanishing results for abelian extension counting series, predicting the exact order of every pole in the stated half-plane. The source presents the preceding theorem and obstruction analysis as evidence, but gives no resolution of the general claim.