Discontinuity conjecture for the Potts model in dimensions at least three

Let $d\ge 3$ and $q\ge 3$. Consider the nearest-neighbor Potts model on $\mathbb{Z}^d$. Discontinuity conjecture. The phase transition of this model is discontinuous. The claim is proved in several regimes, including sufficiently large $q$ at fixed $d$, sufficiently large $d$ at fixed $q$, and sufficiently spread-out interactions, but remains open in general.