Upper combinatorial uniformity conjecture for geometric lattices

A geometric lattice is upper combinatorially uniform when its dual has uniform rank structure, as in the class containing Dowling lattices and perfect matroid designs. A poset is a TN-poset in the sense used in the source's chain-polynomial theory.

Upper combinatorial uniformity conjecture. Every upper combinatorially uniform geometric lattice is a TN-poset.

The source presents this as a proposed extension of the known result for Dowling lattices and notes that perfect matroid designs are also upper combinatorially uniform. No proof or counterexample is supplied.