A longstanding open question in sub-Riemannian geometry is the \(C^\infty \) -smoothness of length-minimizing curves (in their arc-length parameterization). A recent example answered this question negatively, showing the existence of a sub-Riemannian manifold with a length minimizer of class \(C^2\setminus C^3\) . In this paper, we study a broader class of sub-Riemannian structures containing that of the aforementioned example, and we prove that length-minimizing curves are of class \(C^2\) within these structures. In particular, the aforementioned example is sharp (within this class of sub-Riemannian structures).
