Simon's conjecture on primitive groups with near-$2^n$ growth

Let $G$ be a primitive permutation group, and let $f_G(n)$ denote its growth function. Assume that there are polynomials $p(n)$ and $q(n)$ such that

$$\frac{2^n}{p(n)}<f_G(n)<\frac{2^n}{q(n)}.$$

Let $S(2)$ be the local order referred to in the source, and let $\operatorname{Aut}(S(2))$ denote its automorphism group.

Simon's near-$2^n$ conjecture. Then $G$ is either $\operatorname{Aut}(S(2))$, or the group of automorphisms and anti-automorphisms of $S(2)$.

The source attributes this conjecture to Simon and presents it as a further step toward classifying exponential growth rates; no resolution is given in the supplied text.