Macpherson's conjecture on the growth constant for primitive oligomorphic groups

Let $G$ be a primitive oligomorphic permutation group acting on a countable infinite set

. For each positive integer $n$, let $f_n$ be the number of $G$-orbits on the unordered subsets of

of size $n$. Macpherson's theorem states that there is a constant $c>1$ such that either $f_n=1$ for every $n$, or

$$f_n\geq \frac{c^n}{p(n)}$$

for some polynomial $p$. Macpherson's conjecture. The constant can be taken to be $c=2$. The value $c=2$ is the largest possible, and local orders are examples expected to realize this bound; proving that the universal constant in Macpherson's theorem can be chosen to equal $2$ would determine the extremal growth rate for primitive oligomorphic groups.