Global monomial-set conjecture for simple Lie algebras

Let $\boldsymbol{d}:\Delta_+\rightarrow\mathbb{N}$ be a degree function for $U(\mathfrak n^-)$, and let $\mathcal{S}_{\operatorname*{gm}}$ be the set of degree functions such that, for every dominant integral weight $\lambda$, the defining ideal $I^{\boldsymbol{d}}(\lambda)$ of $\operatorname{gr}^{\boldsymbol{d}}V(\lambda)$ is monomial. For a reduced expression $\underline{w}_0$ of the longest Weyl-group element, let $\mathcal{D}_{\underline{w}_0}^q$ denote the corresponding quantum degree cone. Global monomial-set conjecture.

1. $\mathcal{S}_{\operatorname*{gm}}\neq\emptyset$ for any simple finite-dimensional Lie algebra $\mathfrak g$.
2. For any simply-laced simple Lie algebra, there exists $\underline{w}_0$ such that $\mathcal{S}_{\operatorname*{gm}}\cap\mathcal{D}_{\underline{w}_0}^q$ is non-empty.

The known cases include types $A_n$, $C_n$, $B_3$, $D_4$ and $G_2$; the conjecture seeks degree functions producing monomial ideals beyond these cases, with the second assertion imposing compatibility with a quantum degree cone.