Short-distance inversion conjecture for even permutations

Let $n\geq 2$ be even, let $S_n$ be the set of permutations of $[n]$, and consider the statistic counting inversions whose positions have distance at most $3$ (Statistic $494$). An involution without fixed points is an involution on the underlying set having no fixed elements.

Short-distance inversion conjecture. For $n\geq 2$ even, the number of inversions of distance at most $3$ (Statistic $494$) exhibits the cyclic sieving phenomenon under involutions without fixed points.

This is the concluding conjecture of the relevant section. Unlike the preceding results, the supplied text gives no verification range or resolution, so the claim remains open.