The asymptotic scarcity conjecture for blockwise simple permutations

For each $n\in\mathbb{N}$, let $W_n$ be the set of permutations of size $n$, let $\operatorname{Simp}_n$ be the set of blockwise simple permutations, and define

$$A_n=W_n\setminus\operatorname{Simp}_n,\qquad R_n=\frac{|A_n|}{|S_n|},$$

where $S_n$ is the symmetric group. Asymptotic scarcity conjecture. The proportion $R_n$ tends to $0$ when $n$ tends to infinity.

Experimental checks, corroborated through $n=23$, support this asymptotic claim; the conjecture predicts that blockwise simple permutations become asymptotically negligible among all permutations.