Unramified zeta integral continuation and functional equation conjecture

Let $\mathcal{S}\to\operatorname{Spec}(\mathcal{O}_k)$ be a proper, regular model of a smooth projective curve over a number field $k$, let $S$ be a set of curves on $\mathcal{S}$ containing finitely many horizontal curves, and let $f$ and $h$ be the functions used to define the two-dimensional unramified zeta integral

$$\zeta(\mathcal{S},S,f,h,|~|^s).$$

Unramified zeta integral conjecture. The zeta integral meromorphically extends to the complex plane and satisfies

$$\zeta(\mathcal{S},S,f,h,|~|^s)=\zeta(\mathcal{S},S,f,h,|~|^{2-s}).$$

This extends the conjecture attributed in the source to Fesenko concerning two-dimensional zeta integrals and their functional equations. The statement asserts both meromorphic continuation and symmetry under $s\mapsto 2-s$; no resolution is supplied here.