Proof of equality between stabilized branching coefficients and Verschiebung coefficients

Prove that the stabilized coefficients sgn_t(μ)b_{λ,μ}(t) agree with the coefficients a_{λ,μ}(t) describing the action of the Verschiebung operator on universal characters.

Background

The paper introduces stabilized branching coefficients b_{λ,μ}(t) in Lemma 5.5 and uses them in the SXP rules for universal characters in Theorem 5.6. Lemma 5.2 gives alternative expressions for the coefficients a_{λ,μ}(t) arising from plethysm and the Verschiebung operator. Although the paper notes that the two coefficient systems agree in the relevant formulas, it explicitly states that a simple proof of this agreement has not been found.