The compact self-adjointness conjecture for local Hecke operators

Let $\mathcal C$ be a smooth projective curve over a local field $F$ of genus at least $2$, let $G$ be a split connected semi-simple group, and let $\operatorname{Bun}$ be the moduli stack of principal $G$-bundles on $\mathcal C$. For a dominant coweight $\lambda$, let $\mathbb T_{\lambda,c}$ be the corresponding Hecke operator on $L^2(\operatorname{Bun})$. The compact self-adjointness conjecture. The operators $\mathbb T_{\lambda,c}$ are bounded, compact and self-adjoint, and their common spectrum is discrete. This is the expected analytic spectral property distinguishing local-field Hecke operators from the finite-field case; the source gives no resolution status.