The Argentine claim faces the problem that (a) the islands are a quite considerable distance from Argentina -- hence the very limited time that Argentine jets, on marginal fuel, could spend trying to fight our Sea Harriers in 1982 -- and (b) the UN places a high value on 'self-determination' and the British Falkland Islanders are the only permanent human population that the islands have ever had. No Argentine was ever born there, and the presence of men from the precursor United Provinces of the Rio de la Plata, ejected by the US Navy in the 1830s, was transitory and not settled. The issue was only ramped up by the Peron dictatorship as a 'patriotic' rallying point. And the islanders, when asked, overwhelmingly wish to remain British, which is not surprising given the disagreeable behaviour of Argentine troops in 1982 and the ongoing uncertainty of Argentine politics. It will also be remembered that the defeat in 1982 was actually good for Argentina, since it meant the end of the military junta and the 'dirty war' against resisters, when the appearance of a Ford Falcon car carrying men in dark glasses on your street meant the junta's security goons were coming to arrest someone, a process that might well end with that someone being thrown, bound and gagged, out of an aircraft over the Atlantic.
One of the aircraft involved in these flights, I think the only one known to survive, has been found in the US, with its flight logs intact, and is to be returned to Argentina as a memorial to the 'disappeared'.