Chow polynomials and chain polynomials of matroids are not log-concave

The Hilbert--Poincaré series of the Chow rings of the toric varieties of matroids, often called the \emph{Chow polynomials}, and the $f$-polynomial of their boundary complexes, called the \emph{chain polynomials}, have been conjectured to only have real zeros. We construct several explicit infinite