Distinct-domination-sets conjecture for fuzzy Potts phases

Let $J>0$, $q\geq 3$, and $r\in \\{1,\dots,q-1\\}$. Consider the fuzzy Potts model on the torus $\mathbb{T}^d$, and let $\gamma_p$ be Bernoulli product measure of density $p$. Define

$$D_+=\\{p\in[0,1]:\nu_{q,J,r}^{\mathbb{Z}^d,+}\geq\gamma_p\\},$$ $$D_-=\\{p\in[0,1]:\nu_{q,J,r}^{\mathbb{Z}^d,-}\geq\gamma_p\\},$$ $$D_0=\\{p\in[0,1]:\nu_{q,J,r}^{\mathbb{Z}^d,0}\geq\gamma_p\\}.$$

Here $\pi_{q,J}^{\mathbb{T}^d,1}$ and $\pi_{q,J}^{\mathbb{T}^d,0}$ are the corresponding Potts Gibbs measures with the indicated boundary conditions. Distinct-domination-sets conjecture. If $\pi_{q,J}^{\mathbb{T}^d,1}\neq\pi_{q,J}^{\mathbb{T}^d,0}$, then $D_+$, $D_-$, and $D_0$ are pairwise different. The conjecture links phase non-uniqueness in the Potts model to distinct stochastic-domination profiles of the fuzzy phases. The source presents this as an open conjecture and does not provide a proof or disproof.