The EZ lattice-generation conjecture

Suppose $c_0=j/k$ with $k$ and $j$ relatively prime positive integers, and $2\leq k\leq n$. Let $\mathbf{C}[\mathfrak{h}]$ be the polynomial representation, and let $E_1\subseteq\cdots\subseteq E_{\lfloor n/k\rfloor}$ and $Z_{p,m}$ be the submodules defined above. EZ lattice-generation conjecture. The lattice of submodules of $\mathbf{C}[\mathfrak{h}]$ is generated by the submodules

$$E_1\subseteq\cdots\subseteq E_{\lfloor n/k\rfloor}$$

together with those $Z_{p,m}$ such that

$$d_0-d_{-p}+\ell m c_0=p.$$

This stronger conjecture would imply the preceding assertion about $Z_{p,m}'$. Both conjectures concern the submodule structure of the polynomial representation, and no resolution is given in the source.