Mermy, What if AJ asked LE not to notify her parents that she had been in contact with them. Would LE just cancel the alerts and let the parents continue to think/say she is missing and continue to ask the public for help and "donations"? I suppose once the alerts were canceled the parents would figure out that she must have been in contact with LE. I would hope LE would at least give a statement to the public saying AJ has been in touch with LE and she is alive and doing well.
They would notify the family that the case was closed. And, that's most likely all they would do.
But, depending on who handles it within the NPD they might state that they had contact from AJ and they determined that she is safe.