Strong ordering conjecture for pattern-avoidance classes

For a permutation $\tau$, let $S_n(\tau)$ denote the set of permutations in $S_n$ avoiding $\tau$. Suppose $\tau,\sigma\in S_4$. Strong ordering conjecture. If

$$|S_k(\tau)|<|S_k(\sigma)|$$

for some $k$, then

$$|S_n(\tau)|<|S_n(\sigma)|$$

for all $n\geq k$. Equivalently, modulo Wilf-equivalence, permutations can be ordered by relative restrictiveness, with $\tau<\sigma$ when $|S_k(\tau)|<|S_k(\sigma)|$ for some $k$. This conjecture proposes a global ordering of pattern-avoidance classes by their enumeration sequences; the later table in the paper exhibits counterexamples in $S_7$, so the conjecture is refuted.