Equality characterization for the maximal Stanley–Wilf limit

A pattern $q$ of length $k$ is a permutation pattern, and let $S_n(q)$ denote the number of permutations of length $n$ avoiding $q$. Define its Stanley–Wilf limit by

$$L(q)=\lim_{n\to\infty}\sqrt[n]{S_n(q)}.$$

A pattern is layered if it is a direct sum of decreasing permutations. The strengthened Arratia conjecture. For every pattern $q$ of length $k$,

$$L(q)\leq (k-1)^2,$$

with equality if and only if $q$ is layered or the reverse of $q$ is layered. The upper bound is known, while the stated if-and-only-if characterization is presented in the source as a conjectural strengthening and is not resolved there.