Compactness conjecture for Hecke operators

Let $G$ be a reductive group, let $X$ be a curve over a local field $F$, and let $\lambda\ne0$. Write $V_G(\lambda)$ for the dense domain on which the Hecke operators are initially defined and $\mathcal H_G$ for the Hilbert space of half-densities. For each $x\in X$, choose an identification $(\Omega_X^{1/2})_x\cong\mathcal C$. Compactness conjecture. The operators $\widehat{H}_\lambda(x):V_G(\lambda)\to\mathcal H_G$ extend to commuting compact normal operators $H_\lambda(x)$ on $\mathcal H_G$, satisfying $H_\lambda(x)^\dagger=H_{-w_0(\lambda)}(x)$ and having trivial common kernel:

$$\bigcap_{\lambda,x}\operatorname{Ker}H_\lambda(x)=\{0\}.$$

This conjecture would give a discrete decomposition of $\mathcal H_G$ into finite-dimensional joint eigenspaces. It is proved in the simplest nontrivial cases, including $G=PGL_2$ and $X=\mathbb P^1$, but is open in general.