Shared boundary tree topping responsibility (Ohio)

Hello! There is a large oak shared boundary tree on both mine and my neighbor’s property. They had the city inspector out to try to prove it is mine and he ruled it is in fact a shared boundary tree. They are pushing me to pay to have it topped and ideally they would like it removed. However, the reason branches are dying is because they had a new driveway poured that damaged the root system (had a professional arborist out to confirm).

Am I responsible for cutting the limbs over their yard? They’re insisting I am but I don’t believe so.

Thanks in advance!