Existence of Stanley–Wilf limits for all permutation classes

Establish that the Stanley–Wilf limit \(\mathrm{gr}(P)=\lim_{n\to\infty}|P_n|^{1/n}\u000a\) exists for every permutation class \(P\), equivalently, prove that \(\limsup_{n\to\infty}|P_n|^{1/n}=\liminf_{n\to\infty}|P_n|^{1/n}\) for every permutation class.

Background

For a permutation class $P$, the Stanley–Wilf limit is defined when the exponential growth-rate sequence $|P_n|^{1/n}$ has a limit. The Marcus–Tardos theorem gives an exponential upper bound for every proper permutation class, but does not establish convergence of the corresponding sequence.

The paper notes that the limit is known to exist for principal classes $\mathrm{Av}(\pi)$, but states the general existence assertion as a conjecture. The results of the paper concern small permutation classes and do not resolve this conjecture for arbitrary permutation classes.