One-arm conjecture for diffusion-limited aggregation in narrow wedges

Let $A_\infty$ be the limiting diffusion-limited aggregation cluster in a wedge, and interpret its arms as the ends of the graph $A_\infty$. One-arm conjecture. There exists $\theta_0\in (0,2\pi)$ such that for any $\theta\in (0,\theta_0)$, $A_\infty$ has only one arm.

The conjecture predicts that sufficiently narrow wedges produce a single-ended DLA cluster. Simulations suggest that $\theta_0$ is smaller than $\pi/4$; the paper's results do not prove the claim because they lack a lower bound on the growth rate.