Generalized cross-product conjecture for Kahn–Saks coefficients

Prove the generalized cross-product inequality for the Kahn–Saks coefficient function of every poset chain: for all admissible exponent vectors \(\alpha,\beta,\gamma\), establish \(F_P(\alpha)F_P(\alpha+\beta+\gamma)\leq F_P(\alpha+\beta)F_P(\alpha+\gamma)\).

Background

The paper proves a weaker inequality with the multiplicative factor $\binom{|\beta|+|\gamma|}{|\beta|}$, derived from the reverse Khovanskii–Teissier inequality for the Kahn–Saks polynomial. The stronger factor-free inequality is identified as the generalized cross-product conjecture. It concerns the numbers of linear extensions of a poset satisfying prescribed consecutive-distance conditions along a fixed chain.