Universal nonabelian mixed Hodge structure conjecture

Let $X$ be a simply connected smooth projective variety. Universal nonabelian mixed Hodge structure conjecture. There is a universal morphism to a simply connected nonabelian mixed Hodge structure

$$X_M\longrightarrow {\cal Y}={\cal MHS}(X)$$

such that, for every nonabelian mixed Hodge structure ${\cal V}$, the induced morphism

$$\underline{\operatorname{Hom}}({\cal Y},{\cal V})\longrightarrow\underline{\operatorname{Hom}}(X_M,{cal V})$$

is an equivalence. Moreover, this representing object specializes to the stated de Rham and Betti representing $n$-stacks, and its homotopy vector spaces satisfy $\pi_i^{\rm nu}({\cal Y})=\pi_i(X_B)\otimes{\bf C}$ with the mixed Hodge structures defined by Morgan and Hain. The conjecture is intended to construct the mixed Hodge structure on the homotopy type of $X$; it is unproved in the paper.