Residue criterion for the primitive height zeta function

In the same quotient setting, let $\fie GK$ denote the set of isomorphism classes of large $G$-fields over $K$, and let $\delta_{F,m}$ be the residue invariant associated with $F$ and $m$. Write $Z_{X(K)^{\operatorname{prim}}}(s)$ for the primitive height zeta function. Residue criterion conjecture. The function $Z_{X(K)^{\operatorname{prim}}}(s)$ has a simple pole at $s=1$ if and only if

$$\sum_{F\in\fie GK}\delta_{F,m}<\infty.$$

When these equivalent conditions hold,

$$\operatorname{Res}_{s=1}Z_{X(K)^{\operatorname{prim}}}(s)=\frac{\sharp N_{S_n}(G)}{\sharp C_{S_n}(G)\cdot\sharp G}\cdot\sum_{F\in\fie GK}\delta_{F,m}.$$

The source introduces this as a reasonable expectation concerning the possible issue of interchanging limits. No resolution is given.