Cut-locus stratification and branching conjecture in Martinet geometry

In the Martinet case, the cut-locus is the set of endpoints where minimizing geodesics cease to be minimizing; in the generic case, the northern hemisphere and the end-point of the abnormal direction refer to the corresponding local geometry of the sub-Riemannian sphere. A set is $C^1$-stratifiable if it admits a stratification by $C^1$ submanifolds. Cut-locus conjecture. (1) In the Martinet case, the cut-locus is $C^1$-stratifiable. (2) In the generic case, the cut-locus has two branches in the northern hemisphere ramifying at the end-point of the abnormal direction. The Martinet cut-locus is known in the discussed model to be $C^1$-stratifiable, while the stated extension to the generic case is conjectural.