what if St. Peter’s were conquered?

Opening this up for discussion: what would happen if the decline in Italy’s native population continues, and Muslim immigrants in Rome turned aggressive and laid siege to St. Peter’s?

We’d still have “the Church”, but with our increasingly Rome-centric outlook on ecclesial matters in the past few centuries (and especially since the technology has made it easy for John Q. Layperson to read what the Pope says), would this be a shock the likes of which would make Vatican II look like changing from Gather Comp to Gather Comp II?