From what I've been able to gather, the inhabitants of the Island heavily favor Brittain over Argentena, or at least they did back in the first Falklands conflict.
No they still do, most are descendants of Brits. They've made it crystal clear that they want the right to self-determination (essentially aligned with Britain) while the Argentines want to govern the land.
The chances this breaks into armed conflict are smaller than minuscule. While Britain lacks the ability to project power, Argentina can barely provide an offensive capability and can not in anyway survive the economic sanctions if they were the aggressor.