I think you are probably fine, better to wait until the end of January like they said. If nothing changes by Jan 31, follow up again and ask them what you need to do to ensure your application goes through. I mailed in my application instead of dropping it off in person, and my account got updated with receipt and approval of the application on Jan 13. I have friends whose applications were received and approved back in December and/or earlier in January. Unfortunately there does not seem to be any consistency or predictability as to the administrative processing.
I feel like this is untrue, because it would take away from the "blind" evaluation that admissions decisions are traditionally supposed to have. I'd check in with them just in case, but I highly doubt that this is accurate.