I wouldn't pay for half the fence but there are a few things you should consider.
First, as PP have said, you don't have to have a fence. If they want a new fence, they can erect one on their side of the boundary.
Second, if there was something on the deed or a covenant that means you have to pay, the neighbour needs to prove this is the case. In the absence of anything, you don't need to maintain a fence.
Third, whilst the seller's info document is legally binding, it's "to their best knowledge" therefore if the seller believed you a responsible for one fence and not another because that's what they were told, it doesn't in itself prove anything. The neighbour needs evidence to the contrary though because the seller's info doc is more evidence than nothing at all.
Lastly, bear in mind that if they can prove it's your responsibility to maintain the fence, and they've pointed out it needs repairing, you might end up on the receiving end of a civil case if they got injured. Obviously only you can assess whether the fence is safe or not. I wouldn't worry too much about this because they're only saying you're half-liable, therefore if they think it's unsafe they should pay for it to be repaired/replaced and then pursue you for half the cost.
TLDR, tell them to shove it.