The scaling-limit conjecture for vertex-transitive graphs with polynomial ball growth

Let $D_n\leq \Delta_n$ be two sequences going to infinity, and let $(X_n)$ be a sequence of vertex-transitive graphs such that the balls of radius $D_n$ satisfy

$$|B(x,D_n)|=O(D_n^q).$$

Scaling-limit conjecture. Then $(X_n,d/\Delta_n)$ has a subsequence converging for the pointed GH-topology to a connected nilpotent Lie group equipped with a Carnot–Carathéodory metric. This conjecture proposes a nilpotent Lie-group description of subsequential scaling limits under polynomial local growth. The supplied text gives no evidence that the conjecture has been resolved.