External DLA radius growth conjecture on integer lattices

Let $cmathcal{E}_n$ be the external DLA cluster on $cmathbb{Z}^d$ started from $cmathcal{E}_0=\{0\}$, and let $r(cmathcal{E}_n)$ denote its radius. External DLA radius-growth conjecture. The rate of growth of the radius is of order $n^{1/d}$, specifically

$$\limsup_{n\to\infty}n^{-1/d}\mathbb{E}[r(\mathcal{E}_n)]=0.$$

The limit-shape and density problems for external DLA in dimensions $d\geq 2$ remain unresolved, and the source places this conjecture among open questions about the model.