The update-stable zero-one conjecture for wired and free spanning forests

Let $G$ be an infinite network, and let $\mathfrak F$ be either the wired or free spanning forest of $G$. A Borel set $\mathscr A\subseteq \{0,1\}^E$ is update-stable if, for every oriented edge $e$ of $G$, the updated forest $U(\mathfrak F,e)$ belongs to $\mathscr A$ if and only if $\mathfrak F$ belongs to $\mathscr A$ almost surely.

Update-stable zero-one conjecture. For every update-stable Borel set $\mathscr A\subseteq \{0,1\}^E$, the probability that $\mathfrak F$ belongs to $\mathscr A$ is either zero or one.

A positive solution would yield an alternative proof that the number of excessive ends of the wired or free uniform spanning forest is nonrandom, and would also imply a positive solution to Question 15.7 of Benjamini, Lyons, Peres, and Schramm.