The conjectured 123τ–321τ symmetry for alternating involutions

Let $AI_n(\rho)$ denote the set of alternating involutions of length $n$ avoiding the pattern $\rho$. Let $\tau$ be any permutation of $\{4,\ldots,m\}$, where $m\geq 4$. The 123τ–321τ symmetry conjecture.

$$|AI_n(123\tau)|=|AI_n(321\tau)|.$$

This asserts that replacing the initial pattern 123 by 321 preserves the avoidance enumeration for alternating involutions, for every permitted suffix pattern $\tau$. The supplied text gives no resolution beyond its presentation as a conjecture.