Rolnick's growth-rate conjecture for Stanley sequences

Let $S(A)=(a_n)$ be a Stanley sequence, meaning the sequence generated greedily from a finite $3$-free set $A\subset\mathbb N_0$. Rolnick's growth-rate conjecture. For all $n$ large enough, one of the following two patterns of growth is satisfied: Type I, for some constant $\alpha$,

$$\alpha/2\le \liminf_{n\to\infty}\frac{a_n}{n^{\log_2(3)}}\le \limsup_{n\to\infty}\frac{a_n}{n^{\log_2(3)}}\le \alpha,$$

or Type II, $a_n=\Theta(n^2/\ln(n))$. These two patterns are intended to distinguish the highly structured and seemingly random growth behaviors of Stanley sequences; the source attributes the conjecture to Rolnick, but the supplied text gives no resolution.