On chain polynomials of geometric lattices

Athanasiadis and Kalampogia-Evangelinou recently conjectured that the chain polynomial of any geometric lattice has only real zeros. We verify this conjecture for families of geometric lattices including perfect matroid designs, Dowling lattices, and for a class of geometric lattices that contains all lattices of flats of paving matroids. We also investigate how the conjecture behaves with respect to certain operations such as direct products, ordinal sums and single-element extensions.