This work gives sufficient conditions for uniqueness in distribution of semimartingale, obliquely reflecting Brownian motion in a nonpolyhedral, piecewise \(\mathcal {C}^2\) cone, with radially constant, Lipschitz continuous direction of reflection on each face. The conditions are shown to be verified by the conjectured diffusion approximation to the workload in a class of bandwidth-sharing networks, thus ensuring that the conjectured limit is uniquely characterized. This is an essential step in proving the diffusion approximation. This uniqueness result is made possible by replacing the Krein–Rutman theorem used by Kwon and Williams (1993) in a smooth cone with the recent reverse ergodic theorem for inhomogeneous, killed Markov chains of Costantini and Kurtz (Ann Appl Probab, 2024. https://doi.org/10.1214/23-AAP2047; Stoch Process Appl 170:104295, 2024. https://doi.org/10.1016/j.spa.2024.104295).
