Phase-transition conjecture for connectedness of the weighted free uniform spanning forest

Let $H$) be a finite connected graph and let $\mathbb{T}^d$ be the $d$-regular tree. For $w>0$, define the edge-weight function $\hat w$ on $G=\mathbb{T}^d\square H$ by assigning weight $1$ to edges within the tree direction and weight $w$ to edges within the $H$-copies. Write $\mathsf{FSF}_w(G)$ for the free uniform spanning forest of $(G,\hat w)$. Phase-transition conjecture. If $\mathsf{FSF}_{w'}$ and $\mathsf{FSF}_{w”}$ are connected for some $w”>w'>0$, then $\mathsf{FSF}_w$ is connected for every $w\in[w',w”]$; the analogous statement holds for disconnectedness. Moreover, there exists $\gamma\in[0,\infty]$ such that $\mathsf{FSF}_w$ has a unique component whenever $w>\gamma$, and infinitely many components whenever $w<\gamma$. The conjecture describes the expected monotone phase transition in the connectedness of the weighted free uniform spanning forest; the supplied text does not indicate whether it has been resolved.