The Hodge realization conjecture for Nori motivic local systems

Assume that $k$ is algebraically closed. For every smooth, connected $k$-variety $X$, let $\mathcal{ML}_{\sigma}(X)$ be the category of Nori motivic local systems and let $\mathsf{VHS}_{\sigma}(X)$ be the category of admissible, graded-polarizable variations of mixed Hodge structure, with Hodge realization functor

$$\iota_{X,\sigma}^{\textup{Hdg}}\colon\mathcal{ML}_{\sigma}(X)\to\mathsf{VHS}_{\sigma}(X).$$

Hodge realization conjecture. The functor $\iota_{X,\sigma}^{\textup{Hdg}}$ is fully faithful, with essential image stable under subquotients. This is a Hodge-type conjecture for Nori motivic local systems; the statement is formulated for every smooth, connected variety over an algebraically closed field, and remains open.