The conjecture that chromatic noncommutative symmetric functions distinguish oriented paths

Let $D_1$ and $D_2$ be non-isomorphic oriented paths. Their chromatic noncommutative symmetric functions are denoted by $\mathbf{X}_{D_1}$ and $\mathbf{X}_{D_2}$, respectively. Oriented-path distinguishability conjecture. Non-isomorphic oriented paths have different chromatic noncommutative symmetric functions, that is, $\mathbf{X}_{D_1}\ne\mathbf{X}_{D_2}$. The authors verify computationally that chromatic noncommutative symmetric functions distinguish non-isomorphic oriented paths with up to $24$ vertices, while the general statement remains open in the supplied text.