Agreement of naive and Deligne Hodge filtrations for character stacks

Let $G$ be a complex reductive group, and let $\mathcal{Loc}_G(S)$ denote the character stack of a surface $S$. Its Borel–Moore homology carries both the naive Hodge filtration arising from de Rham-to-Dolbeault degeneration and the Hodge filtration coming from mixed Hodge structures. Hodge-filtration agreement conjecture. The naive Hodge filtration on the Borel–Moore homology of character stacks for complex reductive $G$ agrees with the Hodge filtration in the sense of mixed Hodge structure. The conjecture asserts that these two filtrations coincide despite the character stacks being generally non-proper and not necessarily smooth; the preceding discussion notes that such agreement does not hold for arbitrary non-smooth or non-proper varieties.