Mean-periodicity conjecture for arithmetic-surface zeta functions

Let $\mathcal{S}\to\operatorname{Spec}(\mathcal{O}_k)$ be a proper, regular model of a smooth projective curve over a number field $k$. For a finite set of finite Galois extensions $\{k_i/k\}$, let

$$h(\mathcal{S},\{k_i\},x)=f(\mathcal{S},\{k_i\},x)-x^{-1}f(\mathcal{S},\{k_i\},x^{-1}),$$

where $f(\mathcal{S},\{k_i\},x)$ is the inverse Mellin transform of the associated zeta function. A function is $\mathcal{X}$-mean-periodic if its translates have non-dense complex span in the relevant Schwartz space $\mathcal{X}=\textbf{S}(\mathbb{R}^{\times}_{+})$. Mean-periodicity conjecture. There exists a finite set of extensions $\{k_i/k\}$ such that $h(\mathcal{S},\{k_i\},x)$ is $\mathfrak{X}$-mean-periodic, where $\mathcal{X}=\textbf{S}(\mathbb{R}^{\times}_{+})$-mean-periodic. By the result quoted in the source, such mean-periodicity is equivalent to meromorphic continuation and the functional equation of the zeta function, up to sign. The conjecture therefore proposes a mean-periodic formulation of the expected analytic properties, but its resolution is not given here.