Goldman's invariant-forms conjecture for compact-group character varieties

Let $G$ be a compact group, let $\pi$ be the surface group of $\Sigma$, and let $\mathsf{Hom}(\pi,G)/G$ be its character variety. Write $\Omega^*(\mathsf{Hom}(\pi,G)/G)$ for the de Rham algebra of measurable differential forms, and let $\omega_\mathbb{B}$ denote the symplectic structures associated with invariant bilinear forms.

Invariant-forms conjecture. The symplectic structures $\omega_\mathbb{B}$ generate the subalgebra of $\Omega^*(\mathsf{Hom}(\pi,G)/G)$ consisting of $\mathsf{Mod}_{\Sigma}$-invariant forms.

This conjecture extends the known ergodicity of the mapping class group action on compact-group character varieties by predicting that all invariant measurable differential forms are generated by the natural symplectic structures. Its resolution is not stated here.