First-order convergence laws for every proper permutation class

Prove that every proper permutation class satisfies a first-order logical convergence law in the Theory of Two Orders: for every first-order \(\mathrm{TOTO}\) sentence \(\Phi\), show that the probability that a uniformly random permutation in \(P_n\) satisfies \(\Phi\) converges as \(n\to\infty\).

Background

A first-order logical convergence law requires the satisfaction probability of every fixed first-order $\mathrm{TOTO}$ sentence to converge under the uniform distribution on permutations of size $n$ from the class. Prior work established such laws for several specific classes, including layered permutations and certain pattern-avoidance classes.

Braunfeld and Kukla proposed the assertion for all proper permutation classes. The present paper proves it for all small permutation classes, thereby resolving the conjecture only within that growth-rate regime; the general proper-class case remains unresolved in the paper.