The asymptotic prevalence conjecture for exponential graph growth

For each positive integer $n$, let $\operatorname{Perm}(n)$ be the number of isomorphism classes of transitive permutation groups of degree at most $n$, and let $\operatorname{Exp}(n)$ be the number of those classes whose groups have exponential graph growth.

Asymptotic prevalence conjecture. Almost all transitive permutation groups have exponential graph growth, in the sense that

$$\lim_{n\rightarrow\infty}\frac{\operatorname{Exp}(n)}{\operatorname{Perm}(n)}=1.$$

The conjecture is motivated by the reported enumeration through degree $47$, where the overwhelming majority of transitive permutation groups were known to have exponential graph growth. It remains open whether this proportion tends to $1$ as the degree bound tends to infinity.