Variance bound for energy differences between incongruent states

Let $H_{\Lambda,J}$ be the finite-volume Hamiltonian in a box $\Lambda$ with $|\Lambda|=L^d$, and let $\sigma^1$ and $\sigma^2$ be chosen as in Theorem~. Write $\operatorname{Var}$ for variance over the couplings. Variance-bound conjecture. The variance bound of Theorem~ extends to the energy difference between these states:

$$\operatorname{Var}\Big(H_{\Lambda,J}(\sigma^1)-H_{\Lambda,J}(\sigma^2)\Big)\le A L^{d-1},$$

where $A>0$ is a constant. This is proposed as a necessary condition relevant to the presence of incongruent states; the cited upper bound is proved for finite-volume free-energy differences, but its extension to restrictions of infinite-volume pure or ground states remains unproved.