C⁰-rigidity of Legendrians and coisotropics via sheaf quantization

We prove that in the standard cosphere bundle, for any contact homeomorphism in the closure of the compactly supported contactomorphism group, when the image of a coisotropic submanifold (not necessarily properly embedded) is smooth, it is still coisotropic. Moreover, when contactomorphisms in the sequence are in the identity component and the image of a Legendrian is smooth, the Maslov data is preserved, and the category of sheaves with singular support on the Legendrian and the microstalk corepresentative are also preserved (and thus so is the wrapped Floer cochains of the linking disks). The main ingredients are the result of Guillermou--Viterbo, a new sheaf quantization result for C⁰-small contactomorphisms (not necessarily in the identity component) different from Guillermou--Kashiwara--Schapira, and continuity of the interleaving distance of sheaves with respect to the Hofer--Shelukhin distance and the C⁰-distance. The appendix contains different arguments for local C⁰-limits and certain Hausdorff limits of Legendrians without appealing to the interleaving distance.