Characteristic-independence conjecture for IC stalks on the zastava compactification

Let $G$ be a reductive group, let $\overline{\mathrm{Bun}}_{N^-}$ be the compactification used in the paper, and let $\mathrm{IC}_{\overline{\mathrm{Bun}}_{N^-}}$ denote its intersection cohomology sheaf. Characteristic-independence conjecture. If $\mathbb{k}$ is either a finite field or a finite extension of $\mathbb{Q}_\ell$, then for every $i\in\mathbb{Z}$ and every closed point $x\in\overline{\mathrm{Bun}}_{N^-}$, the dimension of

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

as a $\mathbb{k}$-vector space does not depend on the characteristic of $\mathbb{k}$. The conjecture is needed for the paper's construction of the Hecke convolution functor and is expected for general reductive $G$; it is proved in the source when $G=\mathrm{SL}_n$, while a proof in greater generality is only anticipated.