Atomic decomposition conjecture for type $A_n$ crystals

Let $n$ and $k$ be as in the construction of the crystal embeddings, let $\mathcal{B}(k\theta)$ be the crystal of highest weight $k\theta$, and let $\psi_j^n$ denote the specified embeddings. For $i\in[1,n-1]$, define

$$\mathcal{A}_i:=\left(\bigcup_{j\ne i}\operatorname{im}(\psi_j^n)\right)^c.$$

An atom in $\mathcal{B}(\lambda)$ is a subset whose weight multiset equals the weight set $N(\mu)$ for some dominant integral weight $\mu\leq\lambda$; it is a big atom when $\mu=\lambda$. Atomic decomposition conjecture. For every $i\in[1,n-1]$, the subset $\mathcal{A}_i$ is a big atom in $\mathcal{B}(k\theta)$.

This conjecture is based on computer calculations and proposes a systematic way to obtain distinct atomic decompositions of type $A_n$ crystals, contributing to the computation of Kostka--Foulkes polynomials through geometric crystal embeddings. Its resolution is not supplied here.