Beagle Conflict War?

Under the Argentinian military junta of 1976-1983, Argentina suffered from corruption, economic catastrophe, and a popukace that hated the regime. To try and rally support, the Junta rolled the dice on the Falklands, hoping Britain would just hand off the seemingly worthless island. As we all know, Britain would defeat the Argentinans in the Falklands War, resulting in the Junta holding elections and losing control. But before they settled on the Falklands, their initial target was islands south of the Beagle Canal controlled by Chile. War was only really avoided by the pope intervening, ending the crisis. What if the pope didn't intervene, and Chile and Argentina went to war? Who would win? What happens to the Junta?
 
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