Discrete nilpotent loci conjecture for isomonodromic sections

Let $\mathcal T_g$ be the Teichmüller space of genus $g\geq 2$ compact Riemann surfaces, with universal family $\mathcal X/\mathcal T_g$. For an isomonodromic section $\sigma:\mathcal T_g\to\mathcal M_{\mathrm{Dol}}(\mathcal X/\mathcal T_g)$, define its nilpotent loci $\mathcal N\subset\mathcal T_g$ to be the union of all maximal irreducible real analytic subvarieties $S_i\subset\mathcal T_g$, $i\in I$, such that $\sigma|_{S_i}$ is always nilpotent. Discrete nilpotent loci conjecture. The family $\{S_i:i\in I\}$ is a discrete set of subvarieties: for every $s\in\mathcal T_g$, there is a Euclidean open neighborhood $U$ of $s$ such that only finitely many indices $i\in I$ satisfy $U\cap S_i\ne\emptyset$. The conjecture concerns the local distribution of points and subvarieties on which an isomonodromically deformed Higgs bundle remains nilpotent; no resolution is stated in the supplied text.