Explicit Finsler, sub-Finsler, and Carnot-group examples have unique shortest paths with Fuller chattering, giving a negative answer to Le Donne's question.
Fuller ”Relay Control Systems Optimized for Various Performance Criteria”, Automatic and Remote Control, Proceedings of the First IF AC Congress, Moscow, 1960
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Finsler and sub-Finsler geodesics with chattering
Explicit Finsler, sub-Finsler, and Carnot-group examples have unique shortest paths with Fuller chattering, giving a negative answer to Le Donne's question.