The classical limit geometric Langlands conjecture

Let $C$ be the curve, let $G$ and ${}^{L}G$ be a pair of Langlands-dual semisimple groups, and let $\operatorname{\boldsymbol{\mathcal{H}iggs}}$ and ${}^{L}\operatorname{\boldsymbol{\mathcal{H}iggs}}$ denote the corresponding Higgs-bundle moduli stacks. Let $\mathbb{W}^{\mu,x}$ be the classical limit tensorization functors and ${}^{L}\mathbb{H}^{\mu,x}$ the classical limit Hecke functors defined using the classical limit Hecke kernel.

The geometric Langlands correspondence. There exists a canonical equivalence of categories

$$\mathfrak{cl}:D_{\operatorname{coh}}(\operatorname{\boldsymbol{\mathcal{H}iggs}},\mathcal{O})\stackrel{\cong}{\longrightarrow}D_{\operatorname{coh}}({}^{L}\operatorname{\boldsymbol{\mathcal{H}iggs}},\mathcal{O}),$$

which intertwines the action of the classical limit tensorization functors $\mathbb{W}^{\mu,x}$ with the action of the classical limit Hecke functors ${}^{L}\mathbb{H}^{\mu,x}$.

This is the proposed classical-limit, or Higgs-bundle, counterpart of the geometric Langlands correspondence. The source motivates it through the classical limit of the constructions, but gives no evidence that the conjecture has been resolved.