Simpson's foliation conjecture for de Rham moduli spaces

Let $M_{\rm dR}(X,r)$ be the de Rham moduli space, and for each index $\alpha$ let $G_\alpha^1$ be a stratum with projection

$$p_\alpha^1\colon G_\alpha^1\allowbreak\to P_\alpha.$$

Assume that the fibers of these projections are Lagrangian. Simpson's foliation conjecture. When varying $\alpha$, these Lagrangian fibers fit together to provide a smooth foliation of $M_{\rm dR}(X,r)$ with each leaf closed. The conjecture is still open; the source notes progress for rank-$2$ parabolic connections on $\mathbb{P}^1$ minus four points.