Structural mechanism enforcing log-concavity

Establish a structural mechanism that forces the log-concavity conjecture for the differential Hilbert series H_{\beta,M}, potentially by proving that the D_\beta-filtration has an associated graded object carrying a Lorentzian or Lefschetz structure.

Background

Beyond stating log-concavity as a conjecture, the paper asks for an explanation of why the property should hold. The relevant filtration is generated by the interaction of the canonical global lowering operator D_\beta with the coordinate-lowering operators L_a\beta on the graded Möbius algebra of a simple matroid.

The authors specifically suggest investigating whether the associated graded object of this filtration possesses a Lorentzian or Lefschetz structure, which could provide a conceptual proof of the observed inequalities. The paper does not establish such a structure.

References

What mechanism forces Conjecture~\ref{conj:lc}? Does the $D_\beta$-filtration have an associated graded object carrying a Lorentzian or Lefschetz structure, perhaps in the sense suggested by recent structural results on Lefschetz modules ?

— The radial derivative on the graded Möbius algebra  (2608.18519 - Sinclair, 19 Aug 2026) in Section 5, Questions

Can front-loaded defect be realized by maps between complementary defect spaces, rather than only as a numerical inequality?

— The radial derivative on the graded Möbius algebra  (2608.18519 - Sinclair, 19 Aug 2026) in Section 5, Questions