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Does "complete" or "complete and under review" status mean necessarily that no more information will be requested from the applicant? Or is there some chance that the school will hold the app and require more information? Thanks.

It generally means that your application is considered complete and that no additional information is needed. The only exception would, of course, be any KJD candidates who are going to be sending updated transcripts as they become available. But that's not information the school needs to start looking over your materials.