Square-root diameter bound for random homomorphism ranges

Let $G_n$ be a family of vertex-transitive bipartite graphs with $|G_n|\to\infty$. Let $\Delta_n$ denote the diameter of $G_n$, and let $f_n$ be uniformly random in $\operatorname{Hom}_{v_n}^0(G_n,\mathbb{Z})$. Square-root diameter conjecture.

$$\mathbf{E}[R(f_n)]=O\left(\sqrt{\Delta_n}\right).$$

The conjecture seeks a typical range substantially smaller than the naive $O(\Delta_n)$ bound for homomorphisms. The source notes that vertex transitivity is essential: without it, a star with $\Delta$ arms of length $\Delta$ has expected range $\Theta(\Delta)$; the general vertex-transitive case remains open.