Characterization of positivity by completely monotone Dirichlet-series products

Let $\zeta(s)$ be a generalized Dirichlet series absolutely convergent on the half-plane $\{z: \operatorname{Re} z\ge 1\}$, and suppose that it has a simple zero at $s=1$. A generalized Dirichlet series has positive coefficients if all its coefficients are positive; a function is completely monotone on $(1,\infty)$ if it has derivatives of every order there and $(-1)^n f^{(n)}(s)\ge 0$ for every integer $n\ge 0$. The conjecture. $\zeta(s)$ is positive on $(1,\infty)$ if and only if there exists a generalized Dirichlet series $\zeta_2(s)\not\equiv 0$ with positive coefficients, absolutely convergent on the half-plane $\{z: \operatorname{Re} z\ge 1\}$, such that

$$\frac{\zeta(s)\zeta_2(s)}{s(s-1)}$$

is completely monotone on $(1,\infty)$. This conjecture extends finite-dimensional characterizations related to trumping to generalized Dirichlet series in infinite dimensions; the supplied text does not establish the equivalence, so its resolution remains open.