Simpson's nestedness conjecture for Dolbeault and de Rham stratifications

Let $M$ be a (quasi-)projective variety with a stratification by locally closed subsets

$$M=\coprod_{\alpha\in\Lambda}G_\alpha.$$

The stratification is nested if there is a partial order $(\Lambda,\leq)$ such that

$$\overline{G_\alpha}=\coprod_{\beta\leq\alpha}G_\beta.$$

For the corresponding stratifications of $M_{\rm Dol}(X,r)$ and $M_{\rm dR}(X,r)$, Simpson's nestedness conjecture. Both stratifications are nested, and the arrangements for both stratifications are the same. Simpson proved this for rank $2$ using deformation theory, while the higher-rank case remains open.