Declive could be right, I could SPECULATE they might have went to or near Lake Titicaca at the Peruvian-Bolivian border, the question is if there is enough time for the sighting in Lisbon to match up with any other flight except at Comodoro Arturo Benitez Int'l Airport. From my guess/speculation, they probably went this route:
Leg 1 (U.S. --> Chile) / Los Angeles to Santiago and bus to Iquique
Leg 2 (Chile --> Peru --> Bolivia) / bus connecting from Arequipa to Lake Titicaca
Leg 3 (Bolivia --> Portugal) / went across the lake and drove to La Paz Int'l Airport to purchase a flight to Lisbon [via Brazil or Spain].
But the biggest question of all is: Was there a sighting of them at a South American airport after arriving in Chile?