Critical-point threshold conjecture for Gibbs-measure uniqueness

Let $d$ be the dimension, let $p_c$ be the critical point, and let ${\bar{\mu}}_{p_c,q}^1$ and ${\bar{\mu}}_{p_c,q}^0$ denote the Gibbs measures obtained with the two boundary conditions indexed by $1$ and $0$, respectively. Let $q$ be the model parameter.

Critical-point threshold conjecture. There is a $q_0=q_0(d)$ such that for $q>q_0$ there is non-uniqueness of Gibbs measures

$${\bar{\mu}}_{p_c,q}^1\neq {\bar{\mu}}_{p_c,q}^0$$

at the critical point, while for $q<q_0$ the Gibbs measures agree, namely

$${\bar{\mu}}_{p_c,q}^1={\bar{\mu}}_{p_c,q}^0.$$

This conjecture predicts a dimension-dependent threshold separating discontinuous and continuous phase transitions at the critical point. The source notes that the analogous general picture is known for the random cluster model in dimension two, but does not establish it for the random conductance model or state a resolution.