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My question is with all the current links that point to www.example.com. Can I leave them as they are or do I need to get all the links to point to www.example.com/example/ now?
<edit reason> Sorry, didn't read carefully. If you are redirecting to a new subdirectory under the same url, pr will pass.
That said, it is best practice to avoid redirects whenever possible, so I would still ask those linking to you to update the links to reflect the change.
Why you are redirecting to a subdirectory in the first place is another thread....