The bounded-coefficient Dirichlet-series pole conjecture

Let $\{f_n\}_{n\in\mathbb{N}}\in \ell^\infty(\mathbb{C})$, and define

$$g_{\{f_n\}_{n\in\mathbb{N}}}(z)=\sum_{n\in\mathbb{N}}\frac{f_n}{n^z}.$$

Bounded-coefficient Dirichlet-series conjecture. Every function in this class has no poles to the right of the critical line $\operatorname{Re}(z)=1/2$.

The source presents this as Conjecture A and then states a purported implication for the zeroes of these functions. The supplied text does not establish the pole claim.