Alternating-permutation equivalence implies Wilf-equivalence

For a pattern $p$, let $A_m(p)$ be the set of alternating permutations of length $m$ that avoid $p$. Patterns $p$ and $q$ are equivalent for alternating permutations of a parity if their avoidance classes have equal cardinalities for every length of that parity, and they are Wilf-equivalent if they are avoided by the same number of permutations of every length. Alternating-equivalence conjecture. If permutations $p$ and $q$ are equivalent for alternating permutations of either parity, then $p$ and $q$ are Wilf-equivalent for all permutations.

The conjecture proposes that either alternating-parity equivalence relation refines ordinary Wilf-equivalence. The paper notes that the implication is suggested by numerical data, but it remains unproved.