The second analytic Langlands conjecture on discreteness of the joint spectrum

Let $G$ be a complex reductive group, let $X$ be a complex curve, and let $\mathcal A=D_G\otimes_{\mathbb C}\overline{D}_G$ act on $\mathcal H=L^2(\operatorname{Bun}_G)$. Denote by $\operatorname{Spec}_{\mathcal A}(\mathcal H)$ the joint spectrum of this commuting algebra.

Second analytic Langlands conjecture. The spectrum $\operatorname{Spec}_{\mathcal A}(\mathcal H)$ is discrete; equivalently, $\mathcal H$ is the completed direct sum of finite-dimensional eigenspaces of $\mathcal A$.

Discreteness would make the analytic Langlands spectral decomposition a genuinely discrete decomposition into joint eigensections. The conjecture is proved in the abelian case and in the simplest non-abelian case considered in the paper, but no general proof is stated.