Hain's conjecture on NMHS structures on relative Malcev completions

Let $X$ be a pointed space with fundamental group $\pi_1(X,x)$, let $\mathcal V$ be a variation of Hodge structure with fibre $\mathcal V_x$ and polarization $\langle\cdot,\cdot\rangle$, and let

$$\rho:\pi_1(X,x)\to \operatorname{Aut}(\mathcal V_x,\langle\cdot,\cdot\rangle)$$

be its monodromy representation. Let $S$ be the Zariski closure of the image of $\rho$, and let $\mathscr G$ be the relative Malcev completion of $\pi_1(X,x)$ with respect to the induced representation into $S$.

Hain's conjecture. The relative completion $\mathscr G$ should carry a nonabelian mixed Hodge structure in general. This structure should be a quotient of $\pi_1^{\mathrm{hodge}}(X,x)$.

Hain proved this assertion when the monodromy representation has Zariski-dense image in the relevant reductive algebraic group. The conjectural extension concerns the case where the target is the Zariski closure of the monodromy image.