Ok, I just wanted to be sure of the facts.
Regretfully, the neighbor is correct.
When neighbors share an existing “boundary fence” (a fence straddling a property
line), they are, by law, "co-owners" of it. Neither party may tear the fence down without the other’s permission, and they are equally responsible for its maintenance
. (If you choose to put a boundary fence up, however, and are the only party who uses it, you’re solely responsible for it.)
I am sorry that I don’t have better news, but please understand that I do have an ethical and professional obligation to provide customers with legally correct answers, even when an answer is not favorable to the customer.