Extend the positive binomial transform to wreath products

Determine whether the cycle-type extraction for a fixed-point-refined Worpitzky kernel of the wreath product $\mathbb Z_r\wr\mathfrak S_n$, with separate variables marking fixed points of each color, factors through a positive binomial transform of the corresponding fixed-point-free gamma-polynomial.

Background

The paper’s positive binomial transform connects the fixed-point-free gamma-polynomial for type B signed involutions with the gamma-polynomials of all fixed cycle-type strata. The authors suggest extending this structural phenomenon beyond the hyperoctahedral group.

The proposed setting is the wreath product $\mathbb Z_r\wr\mathfrak S_n$, where fixed points of each color would be tracked by separate variables. The unresolved issue is whether extracting cycle types in that multicolored setting yields an analogous positive binomial transform.