Square-root width exponent conjecture for uniform spanning trees

Let $t_n$ be a uniformly sampled tree of size approximately $n$ in the square-torus uniform spanning-tree model, and let $w(t_n)$ be its Euclidean width. The UST width exponent conjecture. The rescaled width

$$\frac{w(t_n)}{n^\alpha}$$

converges in distribution for some $\alpha\in[0.50,0.55]$. This is an empirical prediction based on simulations of conditioned uniform spanning-tree components; the source gives no proof or resolution.