Characteristic-independence conjecture for IC stalks on the opposite Borel compactification

Let $G=\mathrm{SL}_n$, let $\overline{\mathrm{Bun}}_{B^-}$ be the compactification of the moduli stack of $B^-$-bundles, and let $\mathrm{IC}_{\overline{\mathrm{Bun}}_{B^-}}$ denote its intersection cohomology sheaf. Characteristic-independence conjecture. For every $i\in\mathbb{Z}$ and every closed point $x\in\overline{\mathrm{Bun}}_{B^-}$, the dimension of

$$\mathrm{H}^i((\mathrm{IC}_{\overline{\mathrm{Bun}}_{B^-}})_x)$$

as a $\mathbb{k}$-vector space does not depend on the characteristic of $\mathbb{k}$. This is the stalkwise form of the characteristic-independence hypothesis, and the surrounding argument proves the claim for $G=\mathrm{SL}_n$ using the stated stratification and comparison with known descriptions of IC stalks.