Atkinson's conjecture on the density of simple permutations

Let $\mathcal{C}$ be a proper permutation class, and let $\operatorname{Si}_n(\mathcal{C})$ be the set of simple permutations in $\mathcal{C}$ of length $n$. Let $\mathcal{C}_n$ be the set of permutations in $\mathcal{C}$ of length $n$.

Atkinson's conjecture.

$$\frac{|\operatorname{Si}_n(\mathcal{C})|}{|\mathcal{C}_n|}\longrightarrow 0\qquad\text{as }n\longrightarrow\infty.$$

The source attributes this conjecture to Atkinson and gives no resolution.