The nilpotent singular-support conjecture for lisse Hecke actions

Let $G$ be a reductive group, let $X$ be a smooth proper curve, and let $\Dmod(\Bun_G)$ carry its Hecke action. Let $\Dmod_{\on{Nilp}}(\Bun_G)$ be the subcategory with nilpotent singular support, and let $(\Dmod(\Bun_G))^{\on{lisse}}$ be the subcategory of lisse objects for this action. Nilpotent singular-support conjecture. The inclusion

$$\Dmod_{\on{Nilp}}(\Bun_G)\subset (\Dmod(\Bun_G))^{\on{lisse}}$$

is an equality. This predicts that lisse Hecke behavior is exactly characterized by nilpotent singular support; the source gives no evidence of a resolution.