The Dolbeault–Betti symplectic resolution conjecture for $G$-Higgs moduli spaces

Let $G$ be a complex connected reductive group, and let $M_{Dol}^{vc}(G)$ be the moduli space of $G$-Higgs bundles with vanishing Chern classes over a compact Riemann surface of genus $g\geq 2$. Let $M_B(G)$ be the corresponding Betti moduli space (character variety). Dolbeault–Betti symplectic resolution conjecture. The moduli space $M_{Dol}^{vc}(G)$ admits a symplectic resolution if and only if the moduli space $M_B(G)$ admits a symplectic resolution. The conjecture asks whether the existence of symplectic resolutions is preserved between the Dolbeault and Betti moduli spaces, despite their different geometric constructions; its status is not determined in the supplied text.