The renormalized affine-flag localization conjecture

Let $Op$ be the space of opers with nilpotent singularity, let $\tN/\cG$ be the nilpotent-cone quotient, and let

$$Op\underset{\tN/\cG}\arrowtimes \bD^f(\fD(\Fl^{\operatorname{aff}}_G)_\crit\operatorname{mod})$$

be the ind-completed categorical base change. Let

$$\Gamma_{\Fl, Op}: Op\underset{\tN/\cG}\arrowtimes \bD^f(\fD(\Fl^{\operatorname{aff}}_G)_\crit\operatorname{mod})\to\bD_{\mathrm{ren}}(\hg_\crit\operatorname{mod}_\nilp)$$

be the extended global-sections functor. Renormalized localization conjecture. This functor is an equivalence of categories and is exact. The paper proves full faithfulness and an equivalence on the $I^0$-equivariant subcategories, leaving essential surjectivity for the ambient categories conjectural.