Tightness of the partition-constrained minimization LP for Potts internal energy

Let $d\ge 3$, $q\ge d+1$, and $\beta>0$. Consider the minimization linear program introduced in the paper, whose variables are the probabilities $p_C$ of local views $C$ and whose constraints include the $q$-partition constraints for all $q$-partitions $S$ of size $d$.

Partition-LP tightness conjecture. The minimization LP is tight and yields, for every $d$-regular graph $G$,

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

The conjecture is based on computations for small values of $d$ at fixed $\beta$ and would establish the lower internal-energy bound from the regular-graph Potts conjecture in the range $d\ge 3$, $q\ge d+1$, $\beta>0$.