The GIII plot is based on two historical events which may or may not have been connected.
The pope who succeeded Paul VI in 1978 lasted one month before he died. He was something of a reformer, which caused a number of people to assert that he had been poisoned by the Cardinals, who did not want him to change anything. The conspiracy theories grew after no autopsy was performed and doctors ruled that he had simply died of a heart attack.
The movie, however, implies that the pope was poisoned (no evidence of that) and also that the pope was a real crusading reformer, when in fact the real JP I was really not doing anything all that radical, though he had made noises about getting the Vatican's account books in order.
The second thread comes from a real scandal in the early '80s. The head of a corrupt Italian bank was found hanging from a bridge in London in 1982. People were tried for the murder years later, but acquitted. His bank was partially owned by the Vatican, and he had been the Vatican's main financier.
Calvi (the banker) was a crook and an embezzler, but connections to the death of JPI are very tenuous.
However, much as I tend to dismiss nearly all conspiracy theories, in Italy they are more likely to be true, and there may well be something to this one.