Falling branches from Lime tree on path between Alton Road and Gaston Lane, South Warnborough

This very large lime tree has many dead branches and they frequently fall into gardens where residents, including children, could be harmed. I have raised this previously with the Parish Council who I believe have contacted Hart but there has been no response. The tree is growing next to the footpath between the Alton Road B3349 and Gaston Lane but previous discussions with Hart have proved inconclusive about ownership. The tree had been trimmed approximately 11 years ago but has had no care since, becoming overgrown and regularly dropping large dead branches. This is a danger to life to both residents and people using the footpath. Can this be looked at to determine who is responsible and to get dead branches removed and some of the bulk managed. It is a beautiful tree but if it gets bulkier then there is a risk it could come down and we would lose this tree forever. It needs some attention, thank you.