Universal parameterized LMO invariant for flat connections on 3-manifolds

Let $W$ be a 3-manifold, let $\mathrm{Char}(W)$ be its character variety, and let $\mathcal{L}\in\mathrm{Char}(W)$ be a flat connection. Existence of a universal parameterized invariant. To each pair $(W,\mathcal{L})$, there should exist a universal parameterized LMO invariant

$$\hat{Z}_{\mathrm{LMO}}(W;\mathcal{L}).$$

This proposed refinement is intended to retain geometric data that the classical invariant may miss, particularly information associated with the Johnson cokernel. The supplied text does not specify a construction or establish existence.