Why don't they just have a referendum on the Falkland Islands?

Neither the UK or Argentina is ever going to budge on the sovereignty' of the Islands, and neither side wants another war. So why not setup a plan for a referendum on the sovereignty of the Islands, which would have the backing of the US, EU, the UN ETC and allow the people who live there the choice on what they want to do with their own homeland? Whether they wish to be annexed into Argentina, remain a UK protectorate, or choose independence. And then both nations recognize the decision of the Islanders themselves. And officially bury the hatchet and close the matter. Possibly giving the Islands the legal right to hold any future referendums if they so choose. I cannot see a problem with this. It seems the most logical and democratic thing to do. Or am i just being silly? Lol