Monotonicity conjecture for fuzzy Potts plus states beyond criticality

Let $q\geq 3$, $r\in \\{1,\dots,q-1\\}$, and consider the fuzzy Potts model on the torus $\mathbb{T}^d$. Denote by $J_c$ the critical value corresponding to non-uniqueness of Gibbs states for the Potts model. Monotonicity conjecture beyond criticality. If $J_c<J_1<J_2$, then

$$\nu_{q,J_1,r}^{\mathbb{T}^d,+}\leq\nu_{q,J_2,r}^{\mathbb{T}^d,+}.$$

This conjecture asserts stochastic monotonicity of the plus fuzzy Potts measures in the non-uniqueness regime above the critical interaction. The source gives no evidence of a resolution, so the claim remains open.