Barnabei–Bonetti–Castronuovo–Silimbani's Motzkin enumeration conjecture for alternating involutions

Let $\mathcal{AI}_{m}(\rho)$ and $\mathcal{RAI}_{m}(\rho)$ denote, respectively, alternating and reverse alternating involutions of size $m$ avoiding the pattern $\rho$, and let $M_n$ be the $n$-th Motzkin number,

$$M_n=\sum_{k=0}^{\lfloor n/2\rfloor}\frac{n!}{k!(k+1)!(n-2k)!}.$$

Barnabei–Bonetti–Castronuovo–Silimbani's conjecture. For positive integers $n$,

$$|\mathcal{AI}_{2n}(1432)|=|\mathcal{AI}_{2n}(3214)|=M_n,$$ $$|\mathcal{AI}_{2n-1}(3214)|=|\mathcal{RAI}_{2n-1}(1432)|=M_n-M_{n-2},$$ $$|\mathcal{RAI}_{2n}(1432)|=|\mathcal{RAI}_{2n}(3214)|=M_n,$$

and

$$|\mathcal{AI}_{2n+1}(1432)|=|\mathcal{RAI}_{2n+1}(3214)|=M_n.$$

These are attributed to Barnabei, Bonetti, Castronuovo, and Silimbani; the supplied text gives no resolution status.