Mass gap conjecture for Potts lattice gauge theory

Let $\nu=\nu(\beta,q,d)$ be Potts lattice gauge theory on $\mathbb{Z}^d$. Let $\sigma_N=[0,1]^2\times\{N\}\times\{0\}$ be a plaquette, and for a plaquette $\sigma$ let $W_\sigma=W_{\partial\sigma}$ be its Wilson loop variable. Define the correlation length $\xi_\beta=\xi_{\beta,q,d}$ by

$$-\frac{1}{\xi_\beta}=\lim_{N\to\infty}\frac{\log\left(\operatorname{Cov}_\nu(W_{\sigma_1},W_{\sigma_N}^{-1})\right)}{N}.$$

Let $\beta_c(q,d)$ be the critical inverse temperature from the area-law/perimeter-law conjecture. Mass gap conjecture. For all $\beta\neq\beta_c(q,d)$,

$$0<\xi_{\beta,q,d}<\infty.$$

The existence of the defining limit follows from reflection positivity, but finiteness and positivity away from the critical point are not established in general. The conjecture asserts exponential decay of correlations away from criticality.