It doesn't even need no war.
A better scenario for Argentina is the Junta waiting a few months. Even a couple of months and the British wouldn't have been able to act immediately due to the southern hemisphere winter, and would have had to wait until November to send a task force, by which time momentum would have been lost, and it might well have been lost in the UN forever.
Wait another year and the 1981 Defence white paper cuts would have taken effect, which might have meant a halved RM force with one of Fearless or Intrepid gone (or even both), Invincible sold to Australia, and Hermes having been decommissioned - so no aircraft carriers at all, and a scaled back amphibious force.
Mostly true - however the Flag raising incident on South Georgia resulted in Britain sending 2 SSNs and letting the fact be known
This resulted in the Junta somewhat in a panic rushing the invasion before either the Winter kicked or the Defense cuts started to take effect.
Waiting for the RN to be downgraded might have taken another year or 2 to take effect and I am not sure that they had long enough.
A better bet for the Junta is to occupy the island and when the UN resolution 502 was passed in the UKs favor and the UK invoke Article 51 of the UN charter (the right to self defense) then at a time before the Task force arrives in the area - the Junta should shrug, get the garrison to have changed all of the street names, have a lowering of the flag with a massed televised sing song, shit everywhere and evacuate the island telling everyone that they have courageously 'tweaked the Lions tale'.
Apparently I have been told the Argentinian People would have lapped that up!