Small-angle growth-rate conjecture for diffusion-limited aggregation in wedges

Let $(A_n)_{n\geq 0}$ be the DLA cluster process in the wedge $W_{\theta_1,\theta_2}$, and define its growth rate by

$$\mathrm{gr}((A_n)_{n\geq 0})=\sup\left\{\beta\geq 1/2:\limsup_{n\rightarrow\infty}\frac{\operatorname{diam}(A_n)}{n^\beta}>0\right\},$$

where $\operatorname{diam}$ denotes Euclidean diameter, and let $\mathbb{P}_{\theta_1,\theta_2}$ be the law of the process. Small-angle growth-rate conjecture. The growth rate is $\mathbb{P}_{\theta_1,\theta_2}$-almost surely a constant and, as $\theta\rightarrow 0$, it converges to $1$.

This conjecture supplies the growth-rate lower bound that would help establish the one-arm conjecture in narrow wedges. The source presents it as a more ambitious conjecture; no proof or resolution is given.