Negative correlation conjecture for uniformly random spanning forests

Let $G$ be a graph, and let $\mathbf{F}$ be a uniformly random forest chosen from all forests of $G$. For edges $e,f\in E(G)$, negative correlation conjecture.

$$\mathbb{P}(e,f\in \mathbf{F})\leq \mathbb{P}(e\in \mathbf{F})\mathbb{P}(f\in \mathbf{F}).$$

This conjecture asks whether edge inclusion events are negatively correlated for a uniformly random spanning forest. The source describes it as a well-known open problem and notes that it implies the regular-graph spanning-forest bound above.