Outline

The ghosts have had to cover up a few of their mistakes, but currently Perseus has not noticed them as the entire concept of their existence seems impossible to it. People, on the other hand, abandon these explanations as the cover-up piles up.
The ghosts have been working a long game, trying to figure out a plan. If they realise the task they have set themselves is unachievable, they may give up. It is even possible they may reveal themselves and be integrated into human society. If, alternatively, they believe they have been discovered, they are likely to make irrational decisions that could kill a lot of people, or worse, work.
The ghosts can communicate with hypnotized people through text, but have been searching for a method of non-visual communication such as speech-to-text or getting someone else to read things when they need to be heard by someone else. As of yet, they've had little success, but they could be driven to make a rash decision.
The imposition of an approximately 1 week time limit could come from Perseus' regular scans. If the ghosts have covered up one more person's death than there are ghosts, they can't disguise themselves as every person who who has gone missing simultaneously. If this occurs, and Perseus becomes aware, the ghosts may panic and lash out. If, alternatively, the people of the Perseus collectively agree that the scans are a breach of privacy, or an extremely similar duplicate is made of one of the missing persons, Perseus may remain none-the-wiser.
None-the-less, a big concern should be the disregard the ghosts continue to have for human life. If these people could somehow prove that communications between humans and ghosts needn't be so hostile, the situation could be de-escalated for a much longer lasting positive outcome.