@Sofia380 That threat would be a violation of the review policy. If they made it on platform, and follow through, the review can be removed.
https://www.airbnb.ca/help/article/2673/airbnbs-review-policy
In situations like this, it’s best to initially let the guest think they got away with their transgression. Smile and wave as they leave, and simply say ‘thanks for staying!’ Document any issues. Immediately leave the guest an honest review, detailing the transgression. Hopefully they leave theirs immediately as well. If they don’t, hold off until they do. Their review will not be retaliatory if they suspect they got away with their shenanigans. Once that’s done, submit your claim.
Airbnb has changed the process and claims don’t have to be submitted before the next guest checks in. We now have 14 days to report damage, which is, unfortunately, still within the review window. Guest will of course refuse to pay, and it will escalate to Airbnb, who will refuse as well, as they most often do. Raise your rates just a bit and start putting the extra aside into a damage fund. Then there is no need to bother with guest or Airbnb.
Look at how you can improve your guest screening process. If I had to guess, there would have been a red flag or two with this guest that you over looked. On Airbnb, prevention is the name of the game. Keeping sketchy guests out is key. Once they are in the door, you will always be behind the 8 ball.