Trivial automorphism conjecture for component graphs of the uniform spanning forest

Let $\mathbb G$ be a $d$-dimensional transitive graph for some $d>8$, let $\mathfrak F$ be its uniform spanning forest, and let $r\geq 1$. Write $\mathcal C_r(\mathfrak F)$ for the corresponding component graph. Trivial automorphism conjecture. Almost surely, $\mathcal C_r(\mathfrak F)$ has no non-trivial automorphisms. Moreover, there does not exist a deterministic graph $G$ such that $\mathcal C_r(\mathfrak F)$ is isomorphic to $G$ with positive probability. This conjecture proposes that these component graphs have substantially less symmetry than is ruled out by the known failure of quasi-transitivity and oligomorphicity in dimensions greater than eight.