Jack differences in the Muirhead semiring

Prove that for any fixed τ₀∈[0,∞], and any two integer partitions λ and μ with λ majorizing μ, the normalized Jack difference P_λ(x;τ₀)/P_λ(1;τ₀) − P_μ(x;τ₀)/P_μ(1;τ₀) lies in the Muirhead semiring ℳ_S(ℚ_{≥0}) (and conversely).

Background

This conjecture strengthens evaluation positivity by asking for a decomposition of normalized Jack differences into nonnegative combinations of Muirhead generators (monomials and Muirhead differences), thereby placing them in the Muirhead semiring. It refines the characterization of majorization beyond pointwise positivity.

The authors show this conjecture in several cases (including all two-part partitions and certain extremal μ), and also demonstrate that replacing the semiring by the cone fails in general, highlighting the sharpness of the semiring formulation.