# Does $\text{Bun}_G$ have the homotopy type of a classifying space in positive characteristic?

In these lecture notes by Jacob Lurie, he identifies the homotopy type of $$\text{Bun}_G$$ with that of a certain classifying space $$B\mathcal{P}_{sm}$$ when the group scheme $$G$$ is over $$\mathbb{C}$$. Does something analogous hold (e.g. for the $$\ell$$-adic homotopy type) when the base field is positive characteristic, i.e. $$\overline{\mathbb{F}}_p$$?