Foliation Conjecture for the oper stratification

Let $\mathbb{S}_\alpha\rightarrow V_\alpha$ be the projections associated with the oper stratification of the de Rham moduli space. Foliation Conjecture. The Lagrangian fibers of these projections fit together into a smooth Lagrangian foliation with closed leaves. The conjecture concerns the geometric structure of the oper stratification; the whole foliation conjecture for the moduli space of rank two connections over the four-punctured projective line was proved by the authors.