Global spectral minimizer conjecture for signed circulants $C_n(1,2)$

Let $C_n(1,2)$ be the circulant graph on even $n$ vertices, let $A_\sigma$ range over its signed adjacency matrices, and define

$$\rho_-(n)=2\sqrt{\cos^2(\pi/n)+\cos^2(2\pi/n)}.$$

The switching classes are coordinatized by triangle fluxes and the Hamilton-cycle holonomy; the class with alternating triangle flux and anti-periodic step-$1$ holonomy $\alpha=-1$ has spectral radius $\rho_-(n)$. Global minimizer conjecture. For every even $n\ge 8$,

$$\min_\sigma\rho(A_\sigma)=\rho_-(n).$$

This is a flux-minimization claim in the spirit of Lieb's flux-phase theorem. Exhaustive enumeration verifies it for $n\in\{8,10,12,14,16,18\}$, but the asserted lower bound for all even $n$ remains open.