Internal Hom conjecture for nonabelian mixed Hodge structures

Let ${\cal V}$ and ${\cal Y}$ be nonabelian mixed Hodge structures, and suppose that ${\cal Y}$ is $1$-connected, meaning that its $\pi_0$ and $\pi_1$ are trivial. Internal Hom conjecture. There is a nonabelian mixed Hodge structure $\underline{\operatorname{Hom}}({\cal Y},{\cal V})$, with natural choices for the Hodge and weight filtrations, functorial in ${\cal V}$ and contravariantly in ${\cal Y}$. This is stated as a subsidiary conjecture and is unproved in the paper.