Brownian-motion lower bound for random walks on self-similar groups

Let $G$ be a self-similar group and let $\mathcal{X}$ be its limit space. Let $W_t$ be Brownian motion on $\mathcal{X}$, let $d(0,W_t)$ denote its displacement from the base point, and let $X_n$ be a simple symmetric random walk on $G$. Brownian-motion comparison conjecture. For any such random walk,

$$\mathbb{E}d(0,W_n)\preceq \mathbb{E}|X_n|_G.$$

This conjecture proposes that the expected displacement of Brownian motion on the limit space gives a lower bound for the escape rate of every simple symmetric random walk on the corresponding self-similar group. The surrounding discussion presents it as a conjectural consequence of the relationship between limit spaces and Schreier graphs; no resolution is given in the source.