Asymptotic integrality conjecture for intersection densities

For each $n\geq 3$, let

$$\mathcal{I}_n:=\{\rho(G)\mid G\text{ is transitive of degree }n\},$$

where $\rho(G)$ is the intersection density of $G$.

Asymptotic integrality conjecture.

$$\frac{\left|\left\{\rho(G)\mid G\text{ is transitive of degree }n\right\}\cap\mathbb{N}\right|}{|\mathcal{I}_n|}\xrightarrow[n\to\infty]{}1.$$

This conjecture is motivated by computer searches using SageMath. It predicts that the proportion of integer values among the intersection densities arising from transitive groups of degree $n$ tends to $1$, but no proof or disproof is given in the supplied text.