Limit shape universality for nested fractal graphs

Let $G$ be a nested fractal graph, and let $o$ be a starting vertex such that every graph-metric ball $B_o(n)$ has spatial symmetry. Consider the four single-source Laplacian growth models: IDLA, rotor-router aggregation, divisible sandpiles, and abelian sandpiles. Nested-fractal universality conjecture. Limit shape universality holds on $G$: the four models have the same limit shape. This is posed as a natural extension of the result for the Sierpinski gasket; the general problem remains open.