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, $s_{(\delta ,D)}$, where $D$ is a connected skew diagram with $n$ boxes and $\delta$ is a permutation in the symmetric group $S_n$. In this paper, we combine these two and classify when two skew Schur functions in noncommuting variables are equal: $s_{(\delta ,D)}=s_{(\tau ,T)}$ such that $D\ne T$ if and only if $D$ is a nonsymmetric ribbon, $T$ is the antipodal rotation of $D$, and $\overline{{\tau }^{-1}\delta }$ is an explicit bijection between two set partitions determined by $D$.