Log-concavity of the differential Hilbert series

Prove that for every simple matroid M of rank r, the differential Hilbert function h_k^\beta(M) associated with the graded Möbius algebra and the lowering operators D_\beta and L_a^\beta is log-concave, namely (h_k^\beta(M))^2\ge h_{k-1}^\beta(M)h_{k+1}^\beta(M) for every 1\le k\le r-1.

Background

The paper associates to each simple matroid M a graded cyclic module generated from the top radial vector v_r by the canonical lowering operator D_\beta and the adjoint atom-multiplication operators L_a^\beta. Its graded dimensions define the Hilbert series H_{\beta,M}(q)=\sum_k h_k^\beta(M)q^k. The authors conjecture that this sequence has a stronger shape property than the Whitney numbers of the second kind, whose log-concavity is known to fail for some matroids.

The conjecture is supported by exact computation for all 950 simple matroids on eight elements and by modular computation on a stratified sample of 400 simple nine-element matroids. The generalized theta family, including Larson’s counterexample to Whitney log-concavity, satisfies the relevant low-degree inequality, but no general structural proof is provided.