Odlyzko–Stanley dichotomy conjecture for Stanley sequence growth

Let $S(A)=(a_n)$ be a Stanley sequence, meaning the greedy sequence generated from a finite 3-free set $A$ of nonnegative integers. Let $\alpha$ denote the constant appearing in the first growth alternative. Odlyzko–Stanley dichotomy conjecture. Asymptotically, one of the following two growth rates is satisfied:

$$\frac{\alpha}{2}\le \lim\inf \frac{a_n}{n^{\log_2 3}}\le \lim\sup \frac{a_n}{n^{\log_2 3}}\le \alpha$$

or

$$a_n=O\left(\frac{n^2}{\log n}\right).$$

Type 1 sequences have the first growth rate, while Type 2 sequences have the second. Type 1 examples are known for rational $\alpha$ with denominators that are powers of three, whereas no Type 2 sequence is known; the conjectural dichotomy therefore remains open.