Looks fairly straighforward from that chronology of Teribus - having been relinquished by the British, the Falklands legally became part of Argentina, with a settlement established, and this was subsequently invaded and occupied by the British, there being no war in progress with Argentina at the time.

But that's past history, and rectifing past injustices is very rarely possible without introducing fresh injustices.

The real claim to ownership lies with the penguins anyway.