Coxeter-element invariance conjecture for antichain toggles on fences

Let $F$ be a fence with underlying set $S$. Let $w,w'$ be any two Coxeter elements of the group of antichain toggles $\mathcal{T}$, and let $W,W'$ be the respective groups they generate. For $y\in S$, let $\chi_y$ denote the indicator function associated with $y$.

Coxeter-element invariance conjecture. Any linear combination of the indicator functions $\chi_y$ for $y\in S$ is $c$-mesic or orbomesic under the action of $W$ if and only if it is, respectively, $c$-mesic or orbomesic under the action of $W'$.

This is presented as the antichain analogue of a preceding rowmotion result. The conjecture concerns invariance of mesicity and orbomesicity under the choice of Coxeter element; no general proof or counterexample is given.