Fixed-point homomesy and orbit-structure conjectures for the complement-rotation Foatic action

Let $\mathfrak{S}_n$ be the symmetric group, and let $\overline{\rho}$ denote the complement-rotation Foatic action on $\mathfrak{S}_n$. For $w\in\mathfrak{S}_n$, let $\operatorname{Fix}(w)$ be the number of fixed points, or 1-cycles, and let an orbit mean an orbit under $\overline{\rho}$. Complement-rotation orbit-structure conjecture. The statistic $\operatorname{Fix}$ is 1-mesic on every orbit; every orbit contains a permutation having $1$ as a fixed point; and there is a natural indicator statistic that is $\frac12$-mesic and accounts for the orbits of even size. The first two properties are supported by computations in the source, while the natural indicator statistic is not specified there; the combined claims remain conjectural.