The smoothness and symplecticity conjecture for Betti moduli spaces

Let $G$ be a reductive group over $\mathbb{C}$, let $\psi$ be a homogeneous element of $\mathfrak{g}((t))$ of slope $\nu$, and let $H$ and $\mathfrak{h}$ be the associated group and Cartan algebra. For $\kappa\in H$, write ${}_{\kappa}\mathcal{M}^{0}_{\operatorname{Bet},\psi}$ for the derived Betti moduli stack and ${}_{\kappa}\mathcal{M}_{\operatorname{Bet},\psi}$ for its analytic variant.

Smoothness and symplecticity conjecture. Let $\kappa\in H$. For $\nu>0$, the derived structure on ${}_{\kappa}\mathcal{M}^{0}_{\operatorname{Bet},\psi}$ is trivial, and ${}_{\kappa}\mathcal{M}^{0}_{\operatorname{Bet},\psi}$ is an algebraic stack smooth over $\mathbb{C}$. The analytic stack ${}_{\kappa}\mathcal{M}_{\operatorname{Bet},\psi}$ is a complex analytic manifold with a canonical symplectic structure.

This conjecture predicts that the Betti moduli spaces attached to homogeneous elements have no nontrivial derived structure in positive slope and carry the expected smooth symplectic geometry. The source gives no resolution status.