Mixed Hodge compatibility of the relative-completion map

Let $M$ be a simply connected compact Kähler manifold, let ${\mathscr M}$ be the moduli space of algebraic structures on the manifold underlying $M$, and let ${\mathcal G}_M$ be the relative completion of $\pi_1({\mathscr M},[M])$ with respect to the monodromy representation. Let $G_M$ be the group associated with the monodromy representation of the mapping class group. Relative-completion morphism conjecture. The homomorphism

$${\mathcal G}_M \to G_M$$

induced by the monodromy representation $\pi_1({\mathscr M},[M]) \to {\Gamma}_M$ is a morphism of mixed Hodge structures. This predicts compatibility between the canonical mixed Hodge structure on the relative completion and the Hodge-theoretic structure conjectured for $G_M$; the source gives no resolution.