Stays in is a commonly used phrase and can have several synonyms, depending on the context of its usage. If you are talking about someone living in a particular place, "remains" or "resides" can work as synonyms. For example, "John stays in New York" can be rewritten as "John resides in New York" or "John remains in New York."
If you are talking about keeping a particular thing secret or confidential, some synonyms are "maintains," "keeps," or "withholds." For instance, "She stays in the loop about the project" can be rephrased as "She maintains her knowledge about the project." Overall, using synonyms for a word like "stays in" can make your writing more engaging and deliver the intended message more effectively.