Spectral decomposition conjecture for Hecke operators on half-densities

Let $C$ be a smooth proper curve over a non-archimedean local field $K$, let $G$ be a split reductive group, and let $H(C)$ be the commutative algebra of Hecke operators acting on the Schwartz space of half-densities

$$\mathcal W={\mathcal S}({\operatorname{Bun}}_G,|\omega|^{1/2}),$$

where ${\operatorname{Bun}}_G=\underline{{\operatorname{Bun}}}_G(K)$. For each homomorphism $s:H(C)\to\mathbb C$, define

$$\mathcal W_s(C)=\{w\in\mathcal W(C)\mid hw=s(h)w\},$$

and let $S(C)$ be the set of such $s$ for which $\mathcal W_s(C)\ne\{0\}$. Spectral decomposition conjecture. The Hecke eigenspaces exhaust and direct-sum decompose the space:

$$\bigoplus_{s\in S(C)}\mathcal W_s(C)=\mathcal W(C).$$

This conjecture asserts a discrete spectral decomposition for the commuting Hecke operators. The supplied text gives no resolution or partial result for this decomposition, so its status remains open.