The high-degree free-boundary state nonequilibrium conjecture

Let $r$ be the degree parameter, let $J$ be the interaction parameter, and let $J_{uniq}(r)$ denote the uniqueness threshold. Let $\rho^{\mathrm{edge}}(\mu^{\mathrm{FB}})$ denote the edge pressure of the free-boundary state $\mu^{\mathrm{FB}}$, and let $\rho_\Sigma(\mu^+)$ denote its sofic pressure relative to a sofic approximation $\Sigma$. A measure is $\Sigma$-nonequilibrium when it is not an equilibrium state for $\Sigma$.

High-degree free-boundary state nonequilibrium conjecture. For every $\varepsilon>0$, for all sufficiently large $r$, if $J\geq(1+\varepsilon)J_{uniq}(r)$, then

$$\rho^{\mathrm{edge}}(\mu^{\mathrm{FB}})<\rho_\Sigma(\mu^+).$$

In particular, $\mu^{\mathrm{FB}}$ is $\Sigma$-nonequilibrium for every $\Sigma$.

This predicts that comparison with the plus state detects nonequilibrium of the free-boundary state under an interaction strength arbitrarily close to the uniqueness threshold in the high-degree limit. The supplied passage gives numerical and methodological motivation, but no proof or resolution is stated.