  This work gives sufficient conditions for uniqueness in law of
semimartingale, obliquely reflecting Brownian motion in a nonpolyhedral,
piecewise ${\cal C}^2$ 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 a key
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
(2024).
