Coxeter-length difference under the Simion–Schmidt map

Let $S_n$ be the set of permutations of $[n]$, let $obreak\operatorname{inv}(\sigma)$ denote Coxeter length, and let the Simion–Schmidt map send a permutation to its image. The associated statistic is the difference between the Coxeter lengths of a permutation and its image.

Coxeter-length difference conjecture. The difference in Coxeter length of a permutation and its image under the Simion–Schmidt map (Statistic $123$) exhibits the cyclic sieving phenomenon under involutions with $2^{n-1}$ fixed points.

This is one of the conjectured instances of the cyclic sieving phenomenon in the paper; it was tested computationally up to $n=10$, but no proof or resolution is supplied.