The Weil abscissa conjecture for procyclic groups

For every non-empty set $S$ of primes, define the procyclic groups

$$H_S=\prod_{p\in S}\mathbb{Z}_p,\qquad G_S=\prod_{p\in S}C_p,$$

where $C_p$ is cyclic of order $p$. Let $\alpha_D(S)$ denote the abscissa of convergence of the partial Riemann zeta function

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

The Weil abscissa conjecture. For every set $S$ of primes,

$$\alpha(G_S)=\alpha(H_S)=\alpha_D(S)+1.$$

The conjecture is motivated by the bounds $\frac{1}{L}(\alpha_D(S)+1)\leq\alpha(G_S)\leq\alpha(H_S)\leq\alpha_D(S)+1$, where $L$ is a Linnik constant. Chowla's conjecture that $L=1+\varepsilon$ is a Linnik constant for every $\varepsilon>0$ would imply this conjecture, but it is currently out of reach.