Neighbour asking to take care of branches hanging over their property

There is a cedar elm tree that sits on the property line, but it's rooted on my side of the property. The tree is in a perfectly healthy state.

My neighbor is asking me to take care of the branches that are close to/touching his house. I've read through posts on this sub, and from what I understand, it's not my responsibility to cut/prune branches hanging over his property.

Just to be sure: Am I legally liable for any property damage caused by branches that extend over his property?

I'm located in Texas. Any insights would be appreciated.