Barnabei–Bonetti–Castronuovo–Silimbani's prefix-exchange conjecture for alternating involutions

Let $I_3=123$ and $J_3=321$. For a permutation pattern $\tau$, prefix exchange over alternating involutions means equality of the avoidance-class cardinalities after adjoining any nonempty direct-sum suffix. Barnabei–Bonetti–Castronuovo–Silimbani's conjecture. For every $n\geq1$ and every nonempty pattern $\tau$,

$$|\mathcal{AI}_n(123\oplus\tau)|=|\mathcal{AI}_n(321\oplus\tau)|.$$

The source attributes this conjecture to Barnabei, Bonetti, Castronuovo, and Silimbani and provides no evidence of resolution.