If there's a hill the UK is prepared to die on this is it, 99.9 percent of Falkland Islanders voted to remain part of the UK in a UN observed open election subsequent to the UK retaining control of the islands. Argentina has never maintained any form of civilian population there in nearly 150 years, they want the oil and fishing rights.
Edit in 2013 another vote was held, 99.8 percent.
https://en.wikipedia.org/wiki/2013_Falkland_Islands_sovereignty_referendum?wprov=sfla1