The irregular Riemann–Hilbert isomorphism conjecture for homogeneous elements

Let $G$ be a reductive group over $\mathbb{C}$, let $\psi\in\mathfrak{g}((t))$ be homogeneous of slope $\nu>0$, let $s\in\mathfrak{h}$, and set $\kappa=\exp(s)\in H$. Let ${}_{s}\mathcal{M}_{\operatorname{dR},\psi}$ be the de Rham moduli stack with residue $s$, and let ${}_{\kappa}\mathcal{M}_{\operatorname{Bet},\psi}$ be the corresponding Betti moduli stack. The enhanced irregular Riemann–Hilbert map is

$$\widetilde{\operatorname{RH}}:{}_{s}\mathcal{M}_{\operatorname{dR},\psi}\longrightarrow {}_{\kappa}\mathcal{M}_{\operatorname{Bet},\psi}.$$

Irregular Riemann–Hilbert isomorphism conjecture. For any reductive $G$ over $\mathbb{C}$ and homogeneous element $\psi$ in $\mathfrak{g}((t))$ of slope $\nu>0$, and any $s\in\mathfrak{h}$ with $\kappa=\exp(s)$, the map $\widetilde{\operatorname{RH}}$ is an analytic isomorphism.

This would identify the de Rham and Betti moduli spaces analytically, extending the irregular non-abelian Hodge correspondence in this homogeneous setting. The source gives no resolution status.