Elementary-equivalence conjecture for component graphs of uniform spanning forests

Let $\mathbb G_1$ and $\mathbb G_2$ be $d$-dimensional transitive graphs, let $r_1,r_2\geq 1$, and let $\mathfrak F_1$ and $\mathfrak F_2$ be the uniform spanning forests of $\mathbb G_1$ and $\mathbb G_2$, respectively. Elementary-equivalence conjecture. Almost surely, the component graphs $\mathcal C_{r_1}(\mathfrak F_1)$ and $\mathcal C_{r_2}(\mathfrak F_2)$ are elementarily equivalent; that is, they satisfy the same set of first-order sentences in the language of graphs. The conjecture asks whether the almost-sure first-order theory of these component graphs depends only on the dimension, rather than on the particular transitive graph.