Hain's Hodge structure from a cotangent-fiber microsheaf

Let $X$ be a simply connected algebraic variety. Hain's bar construction equips the cohomology, and at the chain level a suitable model, of the based loop space $\Omega X$ with a mixed Hodge structure. A cotangent fiber is the object corresponding to a cotangent fiber under the microsheaf description of the cotangent bundle.

Cotangent-fiber Hodge microsheaf conjecture. Hain's Hodge structure is induced by a Hodge microsheaf whose underlying microsheaf corresponds to a cotangent fiber.

The conjecture proposes a geometric realization of Hain's loop-space Hodge structure through Hodge microsheaves; establishing such a realization is the remaining issue.