Thomas's pre-Cambrian–Cambrian coincidence conjecture

Let $W$ be a finite reflection group containing $-1$, let $G$ be its Dynkin diagram, and let $\bar G$ be an orientation. Form the Coxeter element $c$ from the oriented diagram, order the roots as they occur among the inversions of $c^{h/2}$, and let $x_i$ have inversion set consisting of the first $i$ roots. Let the pre-Cambrian lattice associated with $\bar G$ be the minimal quotient of $W$ for which all the $x_i$ are left modular, and let $C(\bar G)$ denote the Cambrian lattice associated with $\bar G$. Pre-Cambrian–Cambrian coincidence conjecture. The pre-Cambrian lattice associated with $\bar G$ coincides with the Cambrian lattice $C(\bar G)$. This conjecture appears in the proposed description of Cambrian lattices beyond types $A$ and $B$, whose structure was not yet understood in the paper.