The conjectured enumeration of alternating involutions avoiding 1432 and 3214

Let $AI_n(\rho)$ and $RAI_n(\rho)$ denote, respectively, the sets of alternating and reverse alternating involutions of length $n$ avoiding the pattern $\rho$, and let $M_n$ be the sequence used in the paper. The 1432–3214 enumeration conjecture.

$$|AI_{2n}(1432)|=|AI_{2n}(3214)|=|RAI_{2n}(1432)|=|RAI_{2n}(3214)|=M_n,$$ $$|AI_{2n+1}(1432)|=|RAI_{2n+1}(3214)|=M_n,$$ $$|AI_{2n-1}(3214)|=|RAI_{2n-1}(1432)|=M_n-M_{n-2}.$$

The paper presents these equalities as conjectures based on numerical evidence, covering the patterns 1432 and 3214.