The police will immediately dust the letter for prints, and try to get DNA off the stamp. Only “Unibomber” Ted Kaczynski was smart enough to use gloves and something moist for the stamp. Most fools don’t.
If there is homeowner DNA on the stamp, it is a fraud. If it is person or persons unknown, then the investigation will continue.
If the letter was just dropped in their mailbox, not mailed, there still might be DNA or fingerprints on it.
Must be an old stamp if it's a lickable.