WI: Pope runs to Austria-Hungary

At least Austro-Hungary could reinvent themselves as The Protector of Catholicism IOTL, ensuring that all Catholic ethnic groups there stick together with the Vienna Papacy acting as the glue.
Didn’t he want to move to Trieste at one point and FJ said no? I don’t think it would be a boost to Austrian prestige, but rather a headache for both the state and church. There could be international innuendos about Austrian interference in church matters. On the flip side the Pope could be this weird alternative court attracting conservative elements around it and annoying the Imperial court.