Three-valued Möbius function for dominant-composition posets

Prove that, for every partition λ with repeated elements, the Möbius function μ_{D_S(λ)} of the poset D_S(λ) of S-dominant compositions takes only the values −1, 0, and 1.

Background

The Möbius function of the poset of S-dominant compositions governs the coefficients in the character formulas and, consequently, in the Cauchy identity for staircase matrices. For regular λ, the paper proves that this function is ±1 when the relevant Bruhat interval is retained entirely in the dominant-composition subposet and is 0 otherwise.

For partitions with repeated elements, the poset can fail to be graded, so the regular-case argument does not apply directly. The authors state that the conjecture has been verified in several specific cases and note that it is essential for determining the coefficients in the staircase-matrix Cauchy identity. A proof would establish that all such coefficients arising from this Möbius inversion remain within the set {−1, 0, 1}.