Conway–Guttman asymptotics conjecture for occurrences of length-four patterns

Let $S_{n,r}(q)$ denote the set of permutations of length $n$ with $r$ copies of the pattern $q$, and let $S_n(q)$ denote the set of permutations of length $n$ avoiding $q$. For a pattern $q$, write $|S_{n,r}(q)|$ and $|S_n(q)|$ for their cardinalities.

Conway–Guttman conjecture. For $q \in \{4321,4312\}$, there is a constant $K$ such that

$$|S_{n,r}(q)| \sim K|S_n(q)|.$$

For $q\in \{4123,3412,4231,4213,3142\}$, there is a constant $K$ such that

$$|S_{n,r}(q)| \sim Kn^r|S_n(q)|.$$

This conjecture concerns the seven effective Wilf classes of patterns of length four and is supported by substantial numerical evidence. The paper proves asymptotic results for some of these patterns, but the stated general asymptotics remain conjectural.