Andersen's scalar differential-operator conjecture for compact-group character varieties

Let $G$ be a compact group, let $\pi$ be the surface group of $\Sigma$, and consider the space $C^\infty(\mathsf{Hom}(\pi,G)/G)$ with its natural $\mathsf{Mod}_{\Sigma}$-action. Let $D$ be a differential operator on this space that commutes with the action.

Andersen's scalar-operator conjecture. If

$$C^\infty(\mathsf{Hom}(\pi,G)/G)\xrightarrow{D} C^\infty(\mathsf{Hom}(\pi,G)/G)$$

commutes with the $\mathsf{Mod}_{\Sigma}$-action, then $D$ is a scalar multiple of the identity operator.

The conjecture is presented as a strengthening of the preceding invariant-forms conjecture. The source notes a vanishing result for first cohomology but does not state a resolution of this differential-operator claim.