Fixed-point homomesy conjecture for the reversal-rotation Foatic action

Let $\mathfrak{S}_n$ be the symmetric group, and let $\tau$ denote the reversal-rotation Foatic action on $\mathfrak{S}_n$. For a permutation $w$, let $\operatorname{Fix}(w)$ be the number of fixed points, or 1-cycles, of $w$. Reversal-rotation fixed-point homomesy conjecture. The statistic $\operatorname{Fix}$ is 1-mesic for the action of $\tau$ on $\mathfrak{S}_n$. The source introduces this as the remaining Foatic action for which fixed-point homomesy is possible and reports data on orbit sizes, but gives no proof.