Intersection-cohomology stalk conjecture for Uhlenbeck flag spaces

Let $\fg$ be a simple finite-dimensional Lie algebra, let $\hg$ be the corresponding untwisted affine Lie algebra, and let $\fM^\alpha$ be the associated Uhlenbeck flag space. Suppose it has a stratum $\fM^\alpha_{\gamma,\Gamma,\fP}$ and let $(\phi,\underline{c},\underline{s})$ be a point of that stratum. Write $\chn_+$ for the standard maximal nilpotent subalgebra of the Langlands dual affine Lie algebra, and let $\fu(\chg)$ be the invariant space defined from the nilpotent radical of the standard maximal parabolic. With the notation $\beta_l$, $n_l$, $d_l$, $k_l$, $m$, $g$, and $|\gamma|$ determined by the stratum, consider the intersection-cohomology complex $\IC(\fM^\alpha)$.

Intersection-cohomology stalk conjecture. Its stalk at $(\phi,\underline{c},\underline{s})$ is isomorphic to

$$\bigotimes_{l=1}^m\left(\oplus_{r\in\BN} \Sym^r(\chn_+)_{\beta_l}[2r]\right)^{\otimes n_l}\otimes\bigotimes_{l=1}^g\left(\oplus_{r\in\BN}\Sym(\fu(\chg))^r_{d_l} [2r]\right)^{\otimes k_l}[2|\gamma|].$$

This proposes a representation-theoretic description of the intersection-cohomology stalks for arbitrary untwisted affine Lie algebras, generalizing the paper's computed formula in the $\widehat{sl}_n$ case.