Containment via Macdonald differences

Establish the Macdonald analogue (over the positivity cone ℚ(q,t)_{≥0} for q,t>1) of Theorem 1.3 (containment via Jack positivity): prove that for integer partitions λ and μ, λ contains μ if and only if the normalized difference P_λ(x+1;q,t)/P_λ(1;q,t) − P_μ(x+1;q,t)/P_μ(1;q,t) is Macdonald-positive in the sense of expansions with nonnegative coefficients in the Macdonald basis.

Background

Theorem 1.3 establishes a containment characterization via Jack positivity for various specializations. This conjecture asks for the precise Macdonald counterpart using the Macdonald cone defined by positivity for q,t>1. Such a result would extend containment-type positivity from Jack polynomials to Macdonald polynomials and unify these frameworks.

It is stated explicitly as a conjectural analogue, indicating the desired equivalence between containment and positivity of normalized Macdonald differences under the Macdonald positivity cone.