FUSF heaviness and branching conjecture for nonunimodular transitive pairs

Let $G$ be a transitive graph and let $\Gamma\subset \operatorname{Aut}(G)$ be a nonunimodular closed subgroup acting transitively on $G$. Let $\operatorname{WUSF}$ and $\operatorname{FUSF}$ denote the wired and free uniform spanning forests of $G$, respectively. Suppose that

$$\operatorname{WUSF}\neq\operatorname{FUSF}.$$

FUSF heaviness and branching conjecture. Each tree in the $\operatorname{FUSF}$ is $\Gamma$-heavy and has branching number bigger than one almost surely.

This conjecture concerns the structure of the free uniform spanning forest when the wired and free forests differ on a nonunimodular transitive graph. The supplied text gives no resolution evidence for this claim.