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Maybe they could leave the existing fence and additionally fence the extra portion. I once bought a rental wreck where the guy was an alcoholic... Turned out he had been literally tossing his bottles in the backyard for years apparently so his wife wouldn't notice all his drinking. Lol! As if?! Tons of broken glass mixed with dirt and garbage. That was a lesson I needed to learn, obviously.
Moral of the story, you are going to have to play it where it lies. Work something out if possible, get used to it if possible. The main concern at this point is not allowing the "spare land that nobody owns" to become a dumping ground. You may have an alcoholic neighbor...