Kim–Yoo's representation-theoretic conjecture for two-row genomic Schur functions

Let $\lambda=(\lambda_1,\lambda_2)\vdash n$ with $\ell(\lambda)=2$, and set $l_\lambda=\max\{\lambda_1,\lambda_2+1\}$. For $l_\lambda\le m\le n$, let $\mathsf{Par}(\lambda;m)$ be the set of partitions defined by the preceding formulas, and let $\mathcal E_{\lambda;m}$, $\mathcal E_\mu$, $\mathbf G_E$, $X_\mu$, and $H_m(0)$ have the meanings used for the associated $0$-Hecke modules. A partition $\{\mathcal E_\mu\mid\mu\in\mathsf{Par}(\lambda;m)\}$ of $\mathcal E_{\lambda;m}$ should exist for every $m$ such that

$$\sum_{E\in\mathcal E_\mu}\operatorname{ch}([\mathbf G_E])=s_\mu$$

for every $\mu\in\mathsf{Par}(\lambda;m)$, and, for each such $\mu$, there should be a total order $\mathcal E_\mu=\{E_1\prec_\mu\cdots\prec_\mu E_{|\mathcal E_\mu|}\}$ and a filtration

$$M^\mu_0=\{0\}\subseteq M^\mu_1\subseteq\cdots\subseteq M^\mu_{|\mathcal E_\mu|}=X_\mu$$

of $H_m(0)$-modules with $\mathbf G_{E_i}\cong M^\mu_i/M^\mu_{i-1}$ for all $1\le i\le|\mathcal E_\mu|$. This is the representation-theoretic interpretation conjectured by Kim and Yoo for the Schur expansion of the two-row genomic Schur function $U_\lambda$; the expansion itself is $U_\lambda=\sum_{l_\lambda\le m\le n}\sum_{\mu\in\mathsf{Par}(\lambda;m)}s_\mu$. The claim is stated as Conjecture 7.1 of Kim and Yoo and is not resolved in the supplied source.