Representation-theoretic decomposition conjecture for genomic Schur functions

Let $\lambda$ be a partition with $\ell(\lambda)\le 2$, and let $\mathcal{E}_{\lambda;m}$, $\mathsf{Par}(\lambda;m)$, $\mathbf{G}_E$, and $X_\mu$ be the sets and $0$-Hecke modules introduced above. For each $l_\lambda\le m\le n$, consider a partition $\{\mathcal{E}_\mu\mid \mu\in\mathsf{Par}(\lambda;m)\}$ of $\mathcal{E}_{\lambda;m}$. Representation-theoretic decomposition conjecture. Such a partition exists and satisfies: for every $\mu\in\mathsf{Par}(\lambda;m)$,

$$\sum_{E\in\mathcal{E}_\mu}\operatorname{ch}([\mathbf{G}_E])=s_\mu,$$

and there are a total order $\prec_\mu$ on $\mathcal{E}_\mu=\{E_1\prec_\mu E_2\prec_\mu\cdots\prec_\mu E_{|\mathcal{E}_\mu|}\}$ and a filtration

$$M_0=\{0\}\subseteq M_1\subseteq M_2\subseteq\cdots\subseteq M_{|\mathcal{E}_\mu|}=X_\mu$$

of $H_m(0)$-modules such that

$$\mathbf{G}_{E_i}\cong M_i/M_{i-1}$$

for all $1\le i\le |\mathcal{E}_\mu|$. This would give a representation-theoretic realization of the Schur expansion of the genomic Schur function in the two-row case, while the indecomposability of the modules $\mathbf{G}_E$ is separately left as a future question.