Categorical non-Abelian Hodge correspondence for moduli stacks of quiver connections

Let $X$ be a smooth projective variety over $k$. Write $X_{dR}$ and $X_{Dol}$ for the de Rham and Dolbeault formal-groupoid objects, and let $\mathscr{M}^{top}_{\bullet}(X_{dR})$ and $\mathscr{M}^{top}_{\bullet}(X_{Dol})$ denote the associated simplicial topological stacks. There should also be a suitable substack

$$\mathscr{M}^{top, nice}(X_{Dol}) \subset \mathscr{M}^{top}_1(X_{Dol})$$

and corresponding simplicial versions.

Categorical non-Abelian Hodge conjecture. There are mappings

$$\mathscr{M}^{top}(X_{dR}) \longrightarrow \mathscr{M}^{top, nice}(X_{Dol}),\qquad \mathscr{M}^{top}_1(X_{dR}) \longrightarrow \mathscr{M}^{top, nice}_1(X_{Dol})$$

that induce a “categorical” equivalence

$$\mathscr{M}^{top}_{\bullet}(X_{dR}) \simeq \mathscr{M}^{top, nice}_{\bullet}(X_{Dol})$$

for some suitable meaning of “categorical” in the context of simplicial objects in topological stacks. The same should hold after replacing the simplicial stacks obtained from ordinary mapping stacks by those obtained using derived mapping stacks.

This is proposed as a categorification of the non-Abelian Hodge correspondence for moduli stacks of quiver connections. The precise meaning of the categorical equivalence and the suitable Dolbeault substack remain to be established; the conjecture is likewise intended to extend to the derived setting.