The cyclic-character formula for charge connected components

Let $R=(R_1,\ldots,R_k)$ be a sequence of rectangles. Suppose the distinct rectangles $R_j$ occur with multiplicities $m_j$, and set

$$d_R=\gcd(m_j).$$

For $i\in[d_R]$, let $\ell_{d_R}^{(i)}$ denote the corresponding cyclic character, and let $s_{R_j}$ be the Schur function associated with $R_j$. The charge-$i$ connected component of the generalized dual equivalence graph $\mathcal{T}(R)$ is assigned its character. Cyclic-character formula conjecture. That character is

$$\ell_{d_R}^{(i)}\left[s_{R_1}^{m_1/d_R}\cdots s_{R_k}^{m_k/d_R}\right].$$

This refines the conjectured coincidence of the three stratifications by giving the character of each charge connected component. The paper proves that the total character is the product of the rectangular Schur functions and that the latter two stratifications coincide, but the displayed componentwise plethystic formula remains conjectural.