Geometric and equivariant local curve element conjecture

Let a geometric local curve element and an equivariant local curve element be local curve elements of the same topological type in the framework of the paper. Geometric–equivariant local curve conjecture. Every geometric local curve element coincides with the equivariant local curve element of the same topological type. The source explicitly notes that this claim is false in the almost complex setting by analysis of Ionel–Parker, so the conjecture is refuted in that setting.