The rigidity conjecture for nonhomogeneous Coxeter connections

Let $\mathcal{F}$ be a $\mathcal{C}$-formal type of slope $r/h$, and let $\mathscr{O}\subset\operatorname{\mathfrak{g}}$ be a nilpotent orbit. Write $\mathscr{O}\succeq\mathscr{O}^r_0$ for the orbit-order relation specified by the preceding conjecture. Rigidity conjecture. There exists a rigid connection with formal type $\mathcal{F}$ and unipotent monodromy determined by $\mathscr{O}$ if and only if $r$ satisfies one of the conditions in the classification theorem: $r=h+1$; $r=1$; or $1<r<h$ and the corresponding root-system divisibility conditions hold. This conjecture predicts that the classification known for homogeneous Coxeter connections extends to arbitrary $\mathcal{C}$-formal types, with the nilpotent orbit constrained by $\mathscr{O}\succeq\mathscr{O}^r_0$.