Jack differences with varying parameter in the Jack semiring

Show that for parameters 0<σ≤τ<∞ and integer partitions λ, μ with λ majorizing μ, the normalized Jack difference P_λ(x;τ)/P_λ(1;τ) − P_μ(x;τ)/P_μ(1;τ) belongs to the Jack semiring ℐ_C^σ(ℚ_{≥0}), i.e., it can be expressed as a nonnegative combination of Jack differences and normalized Jack terms at parameter σ.

Background

This conjecture extends the semiring-positivity framework to compare Jack polynomials across parameters, asking that differences at τ decompose into nonnegative combinations built from the parameter σ. It is stronger than simple evaluation positivity and ties parameter-variation to the majorization order.

The paper proves this conjecture for two variables and for certain structured pairs, and discusses obstructions that occur if parameters are reversed (τ<σ), underscoring the necessity of the σ≤τ condition.