The Stanley-Wilf limit and longest-pattern-avoiding-subsequence constant conjecture

Let ${\mathcal A}$ be a proper pattern avoidance class, let ${\mathcal S}_n$ be the permutations of length $n$, and let $L_{\mathcal A}(\Pi_n)$ be the length of a longest subsequence of a uniformly random permutation $d560_n$ whose pattern belongs to ${\mathcal A}$. Define

$$s_{\mathcal A}=\limsup_{n\to\infty}|{\mathcal S}_n\cap{\mathcal A}|^{1/n}$$

and

$$2\sqrt{c_{\mathcal A}}=\limsup_{n\to\infty}\frac{\mathbf{E}L_{\mathcal A}(\Pi_n)}{\sqrt n}.$$

The Stanley-Wilf and subsequence-constant conjecture. Both limit superiors are actual limits, and

$$c_{\mathcal A}=s_{\mathcal A}.$$

The conjecture is motivated by the known existence of the relevant limits for sum- or difference-closed pattern classes and by the increasing-permutation class, but the paper describes the general evidence as fragmentary.