Call you insurance agent to ask if they would cover T damage.
I had a T who without notice moved 2 pit bull dogs in. This T would just allow the dogs out without a leash to wonder without any supervision. The neighbors dog was almost attack. The neighbor informed me of the dogs problem. When I would visited to make repairs the dogs were not there. I ended up serving the T a lease violation and called animal control who informed me the dogs had attacked another animal in the past. The T had someone else remove the dogs and I ended the Lease ASAP. This T also had bought a Trampoline that I had them remove on one of my visits to the property.
Without coverage, who ever has the deeper pockets usually gets screwed. It could be a simple as a neighbor kid breaking their neck while jumping on a trampoline. For a few hundred dollars. It's worth the cost in my book. that cover