Pairwise negative correlation conjecture for the uniform forest measure

Let $G=(V,E)$ be a finite connected graph, and let $\mathbb{P}_{\mathrm{UF}}$ denote the uniform forest measure on $G$. A probability measure on $\{0,1\}^E$ satisfies pairwise negative correlation (p-NC) if, for all distinct edges $e,f\in E$,

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

Uniform forest p-NC conjecture. The measure $\mathbb{P}_{\mathrm{UF}}$ satisfies the p-NC property.

This is presented as a special case of the random-cluster p-NC conjecture and of the conjectured p-NC property for the arboreal gas model. Its status is not resolved in the supplied text.