Negative association for up-events and edges in uniform spanning trees

Let $G$ be a finite connected graph, let ${\bf T}$ be a uniform spanning tree, let $e$ be an edge, and let $A$ be an up-event, meaning an upwardly closed event in the space of subgraphs of $G$, that ignores $e$, meaning that for every subgraph $H$ one has $H\in A$ if and only if $H\cup e\in A$. Negative-association conjecture.

$${\bf P}(A\text{ and }e\in{\bf T})\leq {\bf P}(A){\bf P}(e\in{\bf T}).$$

This would strengthen the preceding pairwise negative-correlation theorem for edges of a uniform spanning tree. The source explicitly states that the truth or falsity of this assertion is unknown.