Fixed-point homomesy conjecture for the complement-inversion Foatic action

Let $\mathfrak{S}_n$ be the symmetric group, and let $\overline{\gamma}$ denote the complement-inversion Foatic action on $\mathfrak{S}_n$. For a permutation $w$, let $\operatorname{Fix}(w)$ be the number of fixed points, or 1-cycles, of $w$. Complement-inversion fixed-point homomesy conjecture. The statistic $\operatorname{Fix}$ is 1-mesic on every orbit of the action of $\overline{\gamma}$. Computations for $n\leq 11$ found no counterexample, while the source states that the general claim remains unproved.