Known uniqueness results say that an embedded submanifold has a unique smooth structure making it an embedded submanifold with the subspace topology, and immersed submanifolds have a unique smooth structure making them immersed if we have a prior fixed topology.
This seems to imply that there is an instance of some immersed submanifold $S\subseteq M$ with a given topology and smooth structure, where it's possible to make $S$ an immersed submanifold with different smooth structure if we're also allowed to endow it with a different topology.
Is there a known example of such a submanifold? My definition is the stricter one that a smooth manifold is the image of an injective immersion. If no example exists, I wouldn't mind relaxing the definition to the image of just an immersion.