Relative sections conjecture for variations of nonabelian mixed Hodge structures

Let $f:X\to S$ be a projective morphism of smooth varieties, and let $\underline{{\cal U}}$ be a (linearized) pre-vnamhs over $X$. Relative sections conjecture. There is a (linearized) pre-namhs

$$\underline{{\cal G}}=\Gamma(X_M/S_M,\underline{{\cal U}})$$

over $S$, whose objects are the relative section stacks, including

$$(\underline{G}_{Hod},W):=\Gamma^{\rm se}(X_{Hod}/S_{Hod},(\underline{U}_{Hod},W)),$$

and

$$(\underline{G}_{B,\mathbf R},W):=\Gamma(X_{B,\mathbf R}/S_{B,\mathbf R},(\underline{U}_{B,\mathbf R},W)).$$

If $\underline{{\cal U}}$ is a vnamhs over $X$, then $\underline{{\cal G}}$ should be a vnamhs over $S$. This is the variational form of the basic nonabelian cohomology conjecture; the geometricity of the relative section stacks is noted as an unresolved technical issue.