Functional-equation formulation of the Riemann hypothesis for graph spectral zeta functions

Let $h_n(s)$ be defined by

$$h_n(s)=(4\pi)^{s/2}\Gamma(s/2)n^{-s}\left(\zeta_{\mathbb{Z}/n\mathbb{Z}}(s/2)-n\zeta_{\mathbb{Z}}(s/2)\right).$$

Functional-equation conjecture. For every $s\in\mathbb{C}$ with $0<\operatorname{Re}(s)<1$,

$$\lim_{n\rightarrow\infty}\left|\frac{h_n(1-s)}{h_n(s)}\right|=1.$$

This is presented as a formulation of the Riemann hypothesis in terms of the functional behavior of spectral zeta functions of graphs, and as heuristic evidence for the validity of the Riemann hypothesis. The source does not state a resolution of this conjecture.