Gaitsgory–Lurie quantum fundamental local equivalence conjecture

Let $G$ be a reductive group, let $\check{G}$ be its Langlands dual group, let $\kappa$ be a level, and let $\operatorname{FLE}$ be the canonical functor

$$\operatorname{FLE}:\operatorname{Shv}_{\kappa,A_{\operatorname{Ran}}}(\operatorname{Gr}_{G,\rho(\omega_X)})^{LN,\chi}\longrightarrow\Gamma^{\lax}(\operatorname{Ran},\operatorname{Rep}_q(\check{G})).$$

Gaitsgory–Lurie quantum fundamental local equivalence conjecture. The functor $\operatorname{FLE}$ is an equivalence.

This is presented as a stronger expected result needed for the construction of the quantum Langlands functor, extending the known construction of the corresponding functor by Gaitsgory and Hayash. Its status remains open.