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In this article, we give an explicit computation of the order spectral selectors of a pair of C¹-close Legendrian submanifolds belonging to an orderable isotopy class. The C¹-local flatness of the spectral distance and the characterisation of its geodesics are deduced. Another consequence is the C¹-local coincidence of spectral and Shelukhin-Chekanov-Hofer distances. Similar statements are then deduced for several contactomorphism groups.
Allais et al. (Fri,) studied this question.