Generalized Iwahori filtration conjecture for simple Lie algebras

Let $P$) be a weight lattice of a simple Lie algebra $\mathfrak{g}$, let $\Delta=\Delta_+\sqcup\Delta_-$ be its set of roots, and let ${\mathbf I}$ be the Iwahori subgroup inside $G[z]$, where $G$ is the Lie group of $\mathfrak{g}$. For each $\lambda\in P$, let $\mathbb{D}_\lambda$ and $\mathbb{U}_\lambda^o$ be the generalized global Weyl modules appearing in the construction, with highest-weight algebra $\mathcal{A}^D_\lambda$; write $\vee$ for the restricted dual.

Generalized Iwahori filtration conjecture. The space of functions $\Bbbk[\mathbf I]$ admits a filtration whose associated graded space is isomorphic to

$$\bigoplus_{\lambda\in P}\left(\mathbb{D}_\lambda\otimes_{\mathcal{A}^D_\lambda}\mathbb{U}_\lambda^o\right)^\vee.$$

The conjecture extends the known filtration result for $\mathfrak{sl}_n$ to arbitrary simple Lie algebras. The authors state that the relevant modules have a free action of the highest-weight algebra, but the asserted filtration and associated-graded description remain unproved because of technical difficulties.