Square DLA exponent conjecture

Let ${\sf TDLA}_0(n,[0,+\infty)^2)$ be the square DLA tree with $n$ vertices, rooted at the corner ${\bf c}$, and let ${\bf v}$ be a uniformly sampled vertex of the tree. Write $w(t_n)$ for its Euclidean width and $d_{t_n}({\bf c},{\bf v})$ for the graph distance from the root corner to ${\bf v}$. The square DLA exponent conjecture. Both

$$\frac{w(t_n)}{n^\alpha}\qquad\text{and}\qquad\frac{d_{t_n}({\bf c},{\bf v})}{n^\beta}$$

converge in distribution when $\alpha=\beta$, for some $\alpha\in[0.55,0.61]$. The claim is an empirically motivated prediction for the common geometric and intrinsic scaling exponent of square DLA; the source reports simulations and no proof or resolution.