Gal's conjecture on gamma-nonnegativity of flag homology spheres

Determine whether the h-polynomial of every flag homology sphere is gamma-nonnegative, as asserted by Gal's conjecture.

Background

The paper introduces gamma-nonnegativity for palindromic polynomials and notes that it implies unimodality of the coefficient sequence. In this context, the authors identify Gal's conjecture as a prominent unresolved problem asking whether gamma-nonnegativity holds universally for h-polynomials of flag homology spheres.

This conjecture is presented as background concerning the broader theory of gamma-nonnegativity rather than as a problem addressed by the paper, whose main results instead concern equivariant gamma-effectiveness of order polytopes of graded posets.

References

A very intriguing problem regarding $\gamma$-nonnegativity is Gal's conjecture: is the $h$-polynomial of a flag homology sphere always $\gamma$-nonnegative?

— Order polytopes of graded posets are gamma-effective  (2505.07623 - D'Alì et al., 12 May 2025) in Section 1, subsection “Ehrhart theory, γ-nonnegativity, and their equivariant counterparts”