Extremal image-size conjecture for pop-stack operators on type A Cambrian lattices

Let $A_n$ be the type-$A_n$ Coxeter system, and let $c$ be any Coxeter element of $A_n$. Write $c^{\to}=s_1s_2\cdots s_n$ for the linear Coxeter element and let $c^{\times}$ be a bipartite Coxeter element. For a Coxeter element $d$, let $\mathrm{Camb}_d$ denote the corresponding Cambrian lattice and let $\mathrm{pop}^{\downarrow}_{\mathrm{Camb}_d}$ be its pop-stack operator.

Extremal image-size conjecture. For every Coxeter element $c$ of $A_n$, we have

$$\left|\mathrm{pop}^{\downarrow}_{\mathrm{Camb}_{c^\to}}(\mathrm{Camb}_{c^\to})\right|\leq\left|\mathrm{pop}^{\downarrow}_{\mathrm{Camb}_{c}}(\mathrm{Camb}_{c})\right|\leq\left|\mathrm{pop}^{\downarrow}_{\mathrm{Camb}_{c^\times}}(\mathrm{Camb}_{c^\times})\right|.$$

The conjecture formalizes numerical evidence that the linear and bipartite Coxeter elements are extremal for the sizes of the images of pop-stack operators. The first inequality is consistent with the known comparison between the linear and bipartite cases, while the general assertion remains open.