Advice For Next Steps In Neighbor Dispute?

Location: Columbus, Ohio

Five months ago, a tree snapped in half on our property and fell across the fence separating our backyard and our neighbors front yard. After taking some time to compare prices on tree removal companies, we decided to take care of the tree ourselves, we hired a company to come remove the remaining brush and we were able to bend the chainlink fence back to a functional state.

Aside from stopping by their house to apologize for the wait while we dealt with the tree, the only initial conversations we had with the neighbor was when he came out thanking God that the tree fell in a way that avoided both our homes.

Then, a couple months later, our neighbor mentions that they think they have some damage to their roof — no mention that they expect us to cover it, no contact from their insurance company.

Now, it's five months later. They've reached out angry that the fence hasnt been fully repaired (fair enough, we'll get it fixed), angry that the remains of the tree are still standing (what is left is still alive and we have no plans to remove it at this moment) and VERY angry that we haven't paid for the $1300 roof repair they had done on their house.

Until now, we had NO idea that there was any ill will, mostly because we had no idea they believe our tree damaged their roof. There was no visible damage at the time or else we would've included that in our first conversations with our insurance company.

They expect us to just give them the money to cover the roofing work they had done and have provided no proof to us that it was the tree that caused the need for repairs.

We've spoken to some lawyer friends who advised that the burden of proof is on them and that we could drop communication with them for now. However, our neighbors just texted over an invoice to prove the work was done (still without any detail of WHAT was done or proof that it was the tree that necessitated the repair) and tensions feel higher than ever. We want to do what is right, but don't want to blindly pay out over $1k for something that may not even involve us. We're unsure of next steps and haven't responded to the most recent text.

What should our next steps be?

ETA: Since it has come up, the fence is fully on our property. We've since looked it up and it is indeed a city ordinance to repair bent fences so we do intend to fix it out of our pocket in a timely manner!