Simpson's VHS characterization conjecture for de Rham strata

For each index $\alpha$, let $G_\alpha^1$ be a de Rham stratum and let $\big(G_\alpha^1\big)^{\rm VHS}\subset G_\alpha^1$ be the subset of polarized complex variations of Hodge structure. Let $G_\alpha^0$ be the corresponding Dolbeault stratum, and let ${\rm NAH}$ denote the non-Abelian Hodge correspondence. Simpson's VHS characterization conjecture.

$$\big(G_\alpha^1\big)^{\rm VHS}=G_\alpha^1\bigcap {\rm NAH}\big(G_\alpha^0\big).$$

The claim characterizes exactly which points of the de Rham stratum arise from the corresponding Dolbeault stratum under non-Abelian Hodge theory; the source does not state whether it has been resolved.