Principal-subspace kernel presentation conjecture for higher-level standard modules

Let $\mathfrak{g}=\mathfrak{sl}(n+1)$, let $\Lambda_i$ be the fundamental weights, and let $\bar{U(\mathfrak{n})}$, $\widetilde{U(\bar{\mathfrak{n}})\bar{\mathfrak{n}}_{+}}$, $\widetilde{I_\Lambda}$, and $\widetilde{f_\Lambda}$ have the meanings fixed in the paper. For $k_0,\dots,k_n,k\in\mathbb{N}$ with $k\geq1$ and $k_0+\dots+k_n=k$, set $\Lambda=k_0\Lambda_0+\dots+k_n\Lambda_n$. Principal-subspace kernel presentation conjecture. In this setting, for every such $\Lambda$,

$$\operatorname{Ker}\widetilde{f_\Lambda}=\widetilde{I_\Lambda}.$$

This is the equivalent completed-algebra formulation of the conjectural presentation of principal subspaces. The paper records corresponding results in several low-rank and level-one cases, but the assertion in the stated generality is not resolved here.