Grimmett–Winkler correlation conjecture for spanning forests

Let $G$ be a (weighted) graph. Let $\mathcal{F}$ be the set of all spanning forests of $G$, and let $\mathcal{B}$ be the probability distribution on spanning forests in which each forest has probability proportional to its weight. For two distinct edges $e_1,e_2$ of $G$, Grimmett–Winkler's spanning-forest correlation conjecture.

$$\mathrm{Pr}_{F\sim\mathcal{B}}[e_1\in F \:|\: e_2\not \in F] \geq \mathrm{Pr}_{F\sim\mathcal{B}}[e_1\in F \:|\: e_2\in F].$$

This is an analogue of the negative correlation inequality for spanning trees. It was stated by Grimmett and Winkler and is still open.