The Schwartz-space and multiplicity-one conjecture for automorphic half-forms

Let $\mathcal C$ be a smooth projective curve over $\mathbb C$, let $G$ be a split connected semi-simple group, and let $\operatorname{Bun}$ be the moduli stack of principal $G$-bundles. Let $\mathcal A=\mathcal D\otimes\overline{\mathcal D}$ act on the Schwartz space $\operatorname{Sch}(\operatorname{Bun})$ of smooth half-forms whose images under all elements of $\mathcal A$ lie in $L^2(\operatorname{Bun})$, and let $\mathcal A^{\mathbb R}$ be the real subalgebra fixed by the stated involution. The Schwartz-space and multiplicity-one conjecture. Every $a\in\mathcal A^{\mathbb R}$ extends to an unbounded self-adjoint operator on $L^2(\operatorname{Bun})$; $\operatorname{Sch}(\operatorname{Bun})$ is stable under all Hecke operators; it equals the image of $\iota_{1/2}$; the actions of $\mathcal A$ and the Hecke operators commute; and it contains a dense invariant subspace that is a direct sum of one-dimensional $\mathcal A$-eigenspaces. The conjecture predicts a global multiplicity-one spectral decomposition, while the source notes partial evidence for $G=\operatorname{PGL}_2$ and related groups.