The L-system parametrization conjecture for real curves

Let $G$ be a connected complex semi-simple group with real structure, let $\mathcal C$ be a compact complex Riemann surface with antiholomorphic involution $\tau$ and no real points, and let ${}^LG={}^LG_s$ be the associated Langlands $L$-group. An L-system is a local system on the non-orientable surface $\mathcal C/\tau$ whose orientation-reversing paths have holonomy in the nontrivial component of ${}^LG$. The L-system parametrization conjecture. The spectrum of Hecke operators on $L^2(\operatorname{Bun})$ is parametrized by L-systems on $\mathcal C/\tau$ with values in ${}^LG$ whose pullbacks to $\mathcal C$ admit the structure of a $G$-oper. This proposes a real-geometric parametrization of the automorphic spectrum; the source gives no resolution status.