Davies–Jenssen–Perkins–Roberts minimisation conjecture for the Potts-model program

Let $G$ be a $d$-regular graph, let $q\ge d+1$, let $\beta>0$, and let $U^q_G(\beta)$ denote the objective associated with the Potts-model linear-programming relaxation. Davies–Jenssen–Perkins–Roberts minimisation conjecture. For $d\ge 3$, the minimisation program has optimum given by $K_{d,d}$, and consequently

$$U^q_{K_{d,d}}(\beta)\le U^q_G(\beta)$$

for every $d$-regular graph $G$. This is the minimisation part of the conjecture from Davies, Jenssen, Perkins and Roberts. The paper establishes the relevant case $d=4$ and notes that the statement is open for $d\ge 5$.