The asymptotic functional-relation conjecture for the discrete torus zeta function

Let $H_n(s)$ be the function defined from the discrete-torus spectral zeta function by the asymptotic expansion

$$H_n(s)=\xi_2(s)+\Omega(s)n^{-2}+O(n^{-4}),$$

where

$$\xi_2(s)=\pi^{-s}\Gamma(s)\zeta(\Delta,s),\qquad \Omega(s)=\frac{1}{3}s\pi^{2-s}\Gamma(s)\zeta(\Delta,s-1).$$

Here $\zeta(\Delta,s)$ is the spectral zeta function of the two-dimensional torus. Asymptotic functional-relation conjecture. For every $s\in\mathbb{C}$ with $\mathrm{Re}(s)\in(0,1)$,

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

The proposition preceding the conjecture proves the corresponding relation without absolute values under the condition $\zeta(\Delta,s)\neq0$; the conjecture asserts that the absolute-value relation continues to hold at zeros. The source gives no resolution status.