Generator-and-relation conjecture for the modules M_\bm

Let $\bm=(m_{i,j})_{1\le i\le j\le n}$ be the multiplicity data defining the module M_\bm, let $\g^a$ be the degenerate Lie algebra acting on it, and let $f_{i,j}$ be the corresponding polynomial variables. Define I_\bm to be the ideal generated by the subspace

$$\U({\mathfrak b})\circ \operatorname{span}\left(f_{i,j}^{\sum_{i\le k\le l\le j}m_{k,l}+1},\ i\le j\right).$$

Generator-and-relation conjecture. The following $\g^a$-module M_\bm is isomorphic to

{\mathbb C}[f_{i,j}]_{i\le j}/I_\bm.

This conjecture extends the corresponding presentation known when \bm is supported on the diagonal. It would provide a uniform generators-and-relations description of the degenerate modules M_\bm.