The last polling had something like 90% of islanders refusing to even talk about a potential referendum. Britain is not the belligerent party here. Britain has offered twice to have the Falklands Islands sovereignty discussed at the International Court of Justice, both times Argentina refused. Argentina abandoned claims to the islands in the 1850 Treaty of Convention (minor historical footnote, it was the US ship the Lexington who removed Argentine settlers from the islands), Argentina then made no further claims on the islands for the next 130 years. Argentina's major claim to the islands is that they inherited the claim from Spain when they gained independence, but at the time of independence the Falkland Islands had been ceded to Britain by Spain for 40 years.
Argentina has a weak claim on the islands based on their geography, but if we base the sovereignty of a country on its distance from a larger, more powerful neighbour then by rights Ireland should still be British, though following that to its natural conclusion, Britain actually belongs to France which belongs to Germany which owns all its neighbours but is still smaller than Russia who should now be considered the only country to exist on the entire Eurasian plate. Oh and the USA? Canada is bigger so it's time to develop a taste for poutine. In short, defining the sovereignty of a country based on it's location is absurd.
It's their land, it's been their land for generations, it's up to them and them alone to determine sovereignty and they're not interested in discussing the issue. They have their own parliament on the islands, they make all their own decisions, raise their own taxes and control their own spending with the exception of the UK military forces on the islands that are only needed because the last time the islands were left undefended, Argentina invaded!
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