DLA growth exponent conjecture for lattice aggregates

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

DLA growth exponent conjecture. There exists $\tfrac{1}{d}<\alpha<1$ such that

$$0<\liminf_{t\to\infty}t^{-\alpha}\operatorname{\mathbb{E}}\operatorname{rad}(A_t)\leq\limsup_{t\to\infty}t^{-\alpha}\operatorname{\mathbb{E}}\operatorname{rad}(A_t)<\infty.$$

The conjecture predicts a nontrivial fractal growth exponent for DLA on $\mathbb{Z}^d$. It is stated as open in the source: no lower bound with an exponent strictly larger than $1/d$ is known, although upper bounds are available from work of Kesten.