Thomas's trimness conjecture for Cambrian lattices

Let $C(\bar G)$ be a Cambrian lattice associated with an oriented Dynkin diagram $\bar G$ of a finite reflection group. A lattice is trim if it has a left modular maximal chain and equally many atoms and join-irreducible elements. Thomas's conjecture. All Cambrian lattices are trim. The Cambrian lattices in types $A$ and $B$ are proved to be trim in the paper; the conjecture concerns the other finite reflection-group types, which were not yet well understood.