Asymptotic ordering conjecture for pattern-avoidance classes

Let $\tau,\sigma\in S_n$, and let $S_k(\tau)$ denote the set of permutations in $S_k$ avoiding $\tau$. Say that $\tau$ is asymptotically smaller than $\sigma$ if $|S_k(\tau)|<|S_k(\sigma)|$ for all sufficiently large $k$. Asymptotic ordering conjecture. Modulo Wilf-equivalence, all permutations in $S_n$ can be ordered asymptotically. This is proposed as a weaker possibility after counterexamples to the stronger ordering claim. The paper provides no resolution in the supplied text.