Weak-excedance mid-points and cycle descents

Let $S_n$ be the set of permutations of $[n]$. Consider the statistic counting weak excedances that are also mid-points of a decreasing subsequence of length $3$ (Statistic $373$), and let the cycle descent number be Statistic $317$.

Weak-excedance mid-point conjecture. The number of weak excedances that are also mid-points of a decreasing subsequence of length $3$ (Statistic $373$) is equidistributed with the cycle descent number (Statistic $317$) and thus exhibits the cyclic sieving phenomenon under involutions with $2^{n-1}$ fixed points.

This is stated among the paper's conjectured cyclic-sieving instances and was tested up to $n=10$; the paper supplies no proof or resolution in the provided material.