The Hecke eigenbasis conjecture for automorphic half-forms

Let $F$ be a non-archimedean or archimedean local field, let $\mathcal C$ be a smooth complete curve over $F$, and let $\operatorname{Bun}$ be the moduli stack of principal $G$-bundles on $\mathcal C$. Let $\mathcal M(\operatorname{Bun})$ be the space of automorphic half-forms and let the Hecke operators act on it. The Hecke eigenbasis conjecture. If $F$ is non-archimedean, $\mathcal M(\operatorname{Bun})$ has a basis of Hecke eigenvectors; if $F$ is archimedean, it has a topological basis of Hecke eigenvectors. The conjecture proposes a spectral decomposition of the automorphic half-form space; the source gives no resolution status.