Pemantle–Steif conjecture on symmetry breaking with plus boundary conditions

Let a Heisenberg model be defined on a graph, with symmetry group $G$. Symmetry breaking (SB) occurs if some Gibbs state has a marginal distribution at a vertex that is not $G$-invariant. Symmetry breaking with plus boundary conditions (SB+) occurs if, for some vertex $v$ and cutsets $C_n\to\infty$, the plus-boundary marginal densities satisfy

$$\inf_n\left\|f_{C_n,v}^{+}-1\right\|_\infty\neq 0.$$

Pemantle–Steif conjecture. For any Heisenberg model on any graph, SB is equivalent to SB+.

The paper proves the analogous equivalence for the rotor model on a tree, but leaves the assertion for arbitrary graphs unproved.