Odlyzko–Stanley conjecture on the growth of $p$-Stanley sequences

Let $p$ be an odd prime. A $p$-Stanley sequence $S_p(A)=a_1,a_2,\ldots$ is generated greedily from a $p$-free set $A$ of nonnegative integers, by adjoining at each step the smallest larger integer that preserves $p$-freeness. Odlyzko–Stanley conjecture. Every $p$-Stanley sequence satisfies one of the following two growth laws:

$$a_n=\Theta\left(n^{\log_{p-1}p}\right)$$

or

$$a_n=\Theta\left(\frac{n^{(p-1)/(p-2)}}{(\log n)^{1/(p-2)}}\right).$$

The first law is the modular, digit-structured behavior, while the second is the conjectured chaotic behavior motivated by the authors' heuristic. The source gives no resolution of this conjecture.