Growth exponent conjecture for one-dimensional long-range diffusion-limited aggregation

Let $R$ be a symmetric random walk on $\mathbb{Z}$ with step distribution satisfying

$$\mathbb{P}(|R_1-R_0|=k) \sim c k^{-1-\alpha}$$

for some $0<\alpha<1$, and let $D_n$ denote the diameter of the aggregate after $n$ particles. Growth exponent conjecture. Then

$$D_n=n^{\beta+o(1)},\qquad \beta=\max\left(2,\frac{1}{\alpha}\right).$$

This conjecture asserts that the lower bound from the cited theorem is sharp in the regime $0<\alpha<1$; the context provided gives no resolution, so the conjecture remains open.