Edge-negative association conjecture for uniform random forests

Let $G=(V,E)$ be a finite graph with no loops or multiple edges. Let $\mathcal F$ be the set of forests in $G$, and let $\Phi$ be chosen uniformly from $\mathcal F$. Say that $\Phi$ is edge-negatively associated when, for distinct edges $e,f\in E$,

$$\mathbb P(e,f\in\Phi)\le\mathbb P(e\in\Phi)\mathbb P(f\in\Phi).$$

The edge-negative association conjecture. For every such graph $G$, the random forest $\Phi$ is edge-negatively associated. This is a longstanding problem concerning negative dependence of the uniform random forest measure. The source describes the displayed property as a relatively weak form of negative dependence and mentions that stronger varieties may also be conjectured, without specifying one precisely.