  We show that if the image of a Legendrian submanifold under a contact
homeomorphism (i.e. a homeomorphism that is a $C^0$-limit of contactomorphisms)
is smooth then it is Legendrian, assuming only positive local lower bounds on
the conformal factors of the approximating contactomorphisms. More generally
the analogous result holds for coisotropic submanifolds in the sense of
arXiv:1306.6367. This is a contact version of the
Humili\`ere-Leclercq-Seyfaddini coisotropic rigidity theorem in $C^0$
symplectic geometry, and the proof adapts the author's recent re-proof of that
result in arXiv:1912.13043 based on a notion of local rigidity of points on
locally closed subsets. We also provide two different flavors of examples
showing that a contact homeomorphism can map a submanifold that is transverse
to the contact structure to one that is smooth and tangent to the contact
structure at a point.
