Whirling homomesy conjecture for m-surjections

Let $[n]=\{1,2,\dots,n\}$ and $[k]=\{1,2,\dots,k\}$. Let $\operatorname{Sur}_m(n,k)$ denote the family of functions from $[n]$ to $[k]$ in which every element of $[k]$ occurs exactly $m$ times, and let $\mathbf{w}$ be the whirling map acting on this family. For $j\in[k]$, define $\eta_j(f)=\#f^{-1}(\{j\})$. Whirling homomesy conjecture. Let $\mathcal{F}=\operatorname{Sur}_m(n,k)$ for $m,n,k\in\mathbb{P}$. Under the action of $\mathbf{w}$ on $\mathcal{F}$, $\eta_j$ is $\frac{n}{k}$-mesic for any $j\in[k]$. This extends the preceding theorem from injections and 1-surjections to general $m$-surjections; the source provides no resolution, so the conjecture remains open.