Even their historical claim is dubious as hell.
For them, it was never about the people.
That doesn't stop the Argentinians from packing their cities with monuments and street names dedicated to the Malvinas Argentinas, and the islands (along with the brie wedge of their Antarctica claim) appear on every national map, right down to the ones on keychain fobs for tourists. That's got to do something to their collective psyche, whether it's saber-rattling season or not.
Right now there's offshore oil exploration taking place in the Falklands - you could see tons of equipment near Stanley. If that comes back lucrative, I'd expect the dispute to heat up accordingly.
I don't think it's a co-incidence given that they think there is vast oil reserves around the Falklands that within a few years the UK will have two of the most powerful aircraft carriers in the world (outside of the US carriers) or that our Type-45 was designed to engage dozens of simultaneous air threats, the argentine military isn't even the threat it was in the early 80's, a British carrier battle group carrying F35's with MBDA Meteors (which are a more advanced but similar missile to American AMRAAMS with a variant having a 200 mile range) with a support group of Type-45's and nuclear subs would make it the most one sided naval battle since The Battle of the Nile in 1798.
Any in-bound Argentine planes would be dead before they got within 300 miles of the Falklands from the F35's not to mention the Type 45's which is acknowledged as the most advanced air defense destroyer in the world at the moment.
I lament that we spend so damn much on the things but within 5-10 years we'll have the 2nd most powerful blue water navy in the World (the question then is do we need that really...)
Throw in that the British government has very effectively used political pressure to block all sales of modern aircraft to the Argentine government they'd be flying an obsolete 40 year old fighter against a state of the art 5th generation fighter, it wouldn't even be a war it'd be a damn blood bath, they can't even afford to run the planes they have now, those planes are a mix of 60's era French Mirages and upgraded vietnam era A4 Skyhawks.
Charming place.
The distinction between islets and islands is the basis of what China is up to with the artificial islands it is constructing. UNCLOS is a really interesting thing to get stuck into if you are a legally inclined geek.