Nested-stratification conjecture for de Rham and Higgs moduli

Let $M_{\rm DR}(X,r)$ and $M_{\rm H}(X,r)$ be the de Rham and Higgs moduli spaces, with the stratifications defined in Proposition; identify each de Rham stratum with the corresponding fixed-point set in $M_{\rm H}(X,r)$. A stratification $M=\coprod_{\alpha}G_{\alpha}$ is nested if there is a partial order satisfying

$$\overline{G}_{\alpha}-G_{\alpha}=\coprod_{\beta<\alpha}G_{\beta}.$$

Nested-stratification conjecture. The stratifications of $M_{\rm DR}(X,r)$ and $M_{\rm H}(X,r)$ are nested, and their strata have the same arrangement under the stated identification. This is a precise formulation of the paper’s proposed closure-order compatibility between the de Rham and Higgs stratifications; it remains open in the supplied text.