Rank-alternating homomesy conjecture for bounded labelings of the type A positive root poset

Let $P=\widetriangle_n$ be the type A positive root poset, with rank function $\operatorname{rk}(p)=0$ for minimal elements, and let $\mathcal{A}^{\ell}(P)$ be the set of labelings under the toggle-promotion operator $\operatorname*{TogPro}$. Define the rank-alternating label sum by

$$\mathcal{R}(\sigma)=\sum_{p\in P}(-1)^{\operatorname{rk}(p)}\sigma(p).$$

Rank-alternating homomesy conjecture. The triple $(\mathcal{A}^{\ell}(\widetriangle_n),\operatorname*{TogPro},\mathcal{R})$ is $0$-mesic when $n$ is even and $\frac{\ell}{2}$-mesic when $n$ is odd.

The authors checked this for $n\leq 6$ and $\ell\leq 3$; the case $\ell=1$ was proved by S. Haddadan. The conjecture concerns homomesy of the rank-alternating statistic under toggle-promotion and remains open in general.