Nestedness Conjecture for the oper stratification

Let $\mathbb{M}_{\mathrm{dR}}(X,r)$ be the de Rham moduli space with oper stratification

$$\mathbb{M}_{\mathrm{dR}}(X,r)=\coprod_{\alpha}\mathbb{S}_\alpha.$$

Nestedness Conjecture. The stratification is nested: there is a partial order on the index set $\{\alpha\}$ such that

$$\overline{\mathbb{S}_\alpha}=\coprod_{\beta\leq \alpha}\mathbb{S}_\beta.$$

The conjecture describes how closures of oper strata are assembled from lower strata. The supplied source context places it among Simpson's oper-stratification conjectures, while the supplied status evidence records a proof of the foliation conjecture rather than a proof of nestedness; its resolution should therefore be checked separately.