Equality of skew Schur functions in noncommuting variables

The question of classifying when two skew Schur functions are equal is a substantial open problem, which remains unsolved for over a century. In 2022, Aliniaeifard, Li and van Willigenburg introduced skew Schur functions in noncommuting variables, s(δ,D), where D is a connected skew diagram with n boxes and δ is a permutation in the symmetric group Sₙ. In this paper, we combine these two and classify when two skew Schur functions in noncommuting variables are equal: s(δ,D) = s(τ,T) such that D≠ T if and only if D is a nonsymmetric ribbon, T is the antipodal rotation of D and τ⁻¹δ is an explicit bijection between two set partitions determined by D.