The problem we're facing is that the validity of the note can't be determined unless the note is acted on, due to the complication provided by the False Seer's existence.

It may be uncouth to say so, but acting on the note really is in our best interests, and this is why - if we execute the guilty, obviously the demons are one down. If we execute an innocent person...that will be both a tragedy and a great loss to the community, the militia in particular. But any further...visions, let's call them, will be uncomplicated. We won't have to doubt the Seer's word, and perhaps more importantly, the Seer won't have to doubt themselves.