Non-ordering conjecture for fuzzy Potts measures on the lattice

Let $q\geq 3$, $r\in \\{1,\dots,q-1\\}$, and consider the fuzzy Potts model on $\mathbb{Z}^d$. For $J>0$, write $\nu_{q,J,r}^{\mathbb{Z}^d,+}$ for its plus-state fuzzy Potts measure, and let stochastic ordering be the usual ordering of probability measures on configurations. Non-ordering conjecture. If $J_1,J_2>0$ with $J_1\neq J_2$, then $\nu_{q,J_1,r}^{\mathbb{Z}^d,+}$ and $\nu_{q,J_2,r}^{\mathbb{Z}^d,+}$ are not stochastically ordered. This conjecture concerns the failure of monotonicity in the interaction parameter; the corresponding statement for the Ising model is proved, while the fuzzy Potts case is presented as open.