The two-type growth conjecture for Stanley sequences

A Stanley sequence $S(A)=\{a_n\}$ is generated by the greedy construction from a finite 3-free set $A$ of nonnegative integers. The two-type growth conjecture. Every Stanley sequence follows one of two patterns of asymptotic growth: Type 1, for which

$$\alpha n^{\log_2 3}/2\le a_n\le \alpha n^{\log_2 3},$$

where $\alpha$ is a constant, or Type 2, for which

$$a_n=\Theta(n^2/\log n).$$

The conjecture formalizes the observed division between highly structured, slowly growing Stanley sequences and chaotic, more rapidly growing sequences. The source gives no resolution or further evidence for this classification.