The 12345–43215 involution-equivalence conjecture

For a pattern $\tau$, let $\mathcal{I}_{n}(\tau)$ denote the number of involutions of length $n$ avoiding $\tau$, and write $\sigma\sim_{\mathcal{I}}\tau$ when the two patterns have equal avoidance counts for every $n$. 12345–43215 equivalence conjecture.

$$12345\sim_{\mathcal{I}}43215.$$

The assertion is one of the two pattern equivalences identified as consistent with the tabulated data for patterns of length five. The source gives no proof or later resolution in the supplied material.