Kionke's conjecture on the Weil abscissa of procyclic groups

Let $S$ be a non-empty subset of the set of prime numbers, and let $H_S=\prod_{p\in S}\mathbb{Z}_p$, where $\mathbb{Z}_p$ is the additive group of $p$-adic integers. Define the shifted partial Riemann zeta function by

$$\zeta_S(s-1):=\prod_{p\in S}\frac{1}{1-p^{-s+1}}.$$

Kionke's conjecture. The shifted partial Riemann zeta function $\zeta_S(s-1)$ has the same abscissa of convergence as the Weil representation zeta function $\zeta^W_{H_S}(s)$. This is one of three equalities predicted by Conjecture A of Kionke for the Weil abscissa of procyclic profinite abelian groups. The supplied text does not state whether this equality has been resolved.