Actually Chile/Argentina is probably the most volatile border in South America - I know Chileans who were serving in the south in the early '80's and they fully expected to be at war at short notice.
The full border was, originally, often in dispute and believe it or not the "honest broker" was the Queen/King of GB.
Back in the 50's and 60's several UK survey companies mapped bits of the border - their maps were used to agree, delimit & demarcate, Senora A has a copy of one on silk - a genuinely DEFINITIVE MAP of several square kms up in the mountains. The Pope brokered the deal over the Beagle Channel - almost all of the border is agreed but there are still a few areas under discussion I think.
The Argentinean claim the the FI won't go away but if they would adopt the same attitude as they now have ot Chile it would be very advantageous to both parties IMHO. Like it or not Argentina is the FI's neighbour and they could do with decent relations. And Argentina could benefit greatly from things such as oil exploration and fishing and supplying food and veg to the FI