Uniqueness of the Gibbs measure away from the critical point

Let $p_c$ denote the critical point of the model, and let $p$ be the model parameter. A Gibbs measure is an infinite-volume Gibbs measure for the random conductance model.

Uniqueness conjecture. For $p\neq p_c$ there is a unique Gibbs measure.

This conjecture proposes that non-uniqueness occurs only at the critical point, strengthening the stated phase-transition and non-uniqueness results away from criticality. The source does not indicate whether it has been proved or disproved.