Dyck-path monomial basis conjecture for the modules M_\bm

Let $\bm=(m_{i,j})$ define M_\bm, let v_\bm be its distinguished cyclic vector, and let $f_\alpha$ denote the commuting generators indexed by positive roots. Define S_\bm\subseteq{\mathbb Z}_{\ge0}^{n(n-1)/2} to consist of collections $\bs=(s_\alpha)_{\alpha>0}$ such that, for every Dyck path $\bp$ starting at $\alpha_i$ and ending at $\alpha_j$,

$$\sum_{\beta\in\bp}s_\beta\le \sum_{i\le k\le l\le j}m_{k,l}.$$

Monomial basis conjecture. The elements

\{f^\bs v_\bm\mid \bs\in S_\bm\}

form a basis of M_\bm. This conjecture proposes an extension of the monomial basis known in the diagonal-support case to arbitrary multiplicity data \bm. It would give an explicit combinatorial basis for M_\bm and connect its dimension and structure with Dyck-path inequalities.