Lewis's even-length alternating-pattern enumeration conjecture

Let $A_n(\sigma)$ be the set of permutations of $[n]$ that avoid the pattern $\sigma$ and are alternating. Lewis's conjecture. For $n\geq 1$ and $\sigma \in \{1243,2134,1432,3214,2341,4123,3421,4312\}$, we have

$$|A_{2n}(\sigma)|=|A_{2n}(1234)|=|A_{2n}(2143)|.$$

These equalities concern the enumeration of even-length alternating permutations avoiding patterns of length four; the common value for the two reference patterns is the $3$-dimensional Catalan number $\frac{2(3n)!}{n!(n+1)!(n+2)!}$. The statement was posed by Lewis and its resolution is not specified in the supplied text.