Lewis's odd-length alternating-pattern enumeration conjecture for 2134, 4312, 3214 and 4123

Let $A_n(\sigma)$ be the set of permutations of $[n]$ that avoid the pattern $\sigma$ and are alternating. Lewis's conjecture. For $n\geq 0$ and $\sigma \in \{2134,4312,3214,4123\}$, we have

$$|A_{2n+1}(\sigma)|=|A_{2n+1}(1234)|.$$

This predicts equal enumerations for odd-length alternating permutations avoiding each of the four specified patterns and avoiding $1234$. The statement was posed by Lewis, and the supplied text does not specify whether it has been resolved.