Strictly superminimal growth of DLA radius on lattices

Let $G=\mathbb{Z}^d$ and let $(A_t)_t$ be DLA on $G$ started at $A_0=\{0\}$. Write $\operatorname{rad}(A_t)$ for the radius of the aggregate.

DLA superminimal growth conjecture.

$$\limsup_{t\to\infty}t^{-1/d}\operatorname{\mathbb{E}}\operatorname{rad}(A_t)=0.$$

This is presented as a naive conjecture and is described as wide open. It would assert that the expected radius grows strictly more slowly than the volume-forced scale $t^{1/d}$.