Pairwise negative correlation conjecture for random-cluster measures

Let $G=(V,E)$ be a finite graph, and let $\phi_{p,q}=\phi_{p,q}(G)$ be the random-cluster measure on $G$ with parameter $q\in(0,1)$. A probability measure on $\{0,1\}^E$ satisfies pairwise negative correlation (p-NC) if, for all distinct edges $e,f\in E$,

$$\mu\big[\omega(e)=\omega(f)=1\big]\leqslant \mu\big[\omega(e)=1\big]\mu\big[\omega(f)=1\big].$$

Random-cluster p-NC conjecture. The measures $\phi_{p,q}$ satisfy the p-NC property.

For $q\in(0,1)$, these measures satisfy the negative lattice condition, but that condition is neither necessary nor sufficient for p-NC. The conjecture proposes a weak form of negative dependence for random-cluster measures in the subunit-$q$ regime.