Nonabelian mixed Hodge structure conjecture for Brill–Noether stacks

Let $X$ be a smooth algebraic curve, let $G$ be an algebraic group, and let $\rho$ be a representation of Hodge type. Write $\kappa(G,\rho,n)$ for the corresponding Brill–Noether stack, and let $\operatorname{Hom}(X,\kappa(G,\rho,n))$ carry the $\mathbb{C}^*$-action induced by the action on differential forms.

Nonabelian mixed Hodge structure conjecture. There is a non-abelian mixed Hodge structure on

$$\operatorname{Hom}(X,\kappa(G,\rho,n)),$$

in the sense of K-P-T, defined by this action.

This conjecture proposes a Hodge-theoretic structure on mapping stacks whose points are described by nonabelian cohomology classes. The source does not state whether the conjecture has been proved or disproved.