Möbius-function values for dominant-composition posets

Prove that, for every partition with repeated elements, the Möbius function of the dominant-composition poset associated with an arborescent poset and its consistent anti-linearization takes only the values −1, 0, and 1.

Background

The paper uses Möbius inversion on posets of dominant compositions to express generalized van der Kallen-module characters and the Cauchy identities for staircase matrices. For regular partitions, the authors derive an explicit criterion showing that the Möbius function takes values in {−1, 0, 1}, with nonzero values equal to signs determined by rank differences.

For partitions with repeated elements, the associated parabolic dominant-composition posets need not be graded. The authors conjecture that the same restricted set of Möbius-function values nevertheless holds. They note that this conjecture is essential for determining coefficients in the staircase-matrix Cauchy identity and for obtaining explicit resolutions of the corresponding minimal subquotients.