Would Proselytizing work on Muslim Europeans?

Hard to imagine that the Catholic Church would be opposed to giving it a shot. Someone by the singular name of Spengler commented in Asia Times Online shortly before Cardinal Ratzinger became the current Pope:

Now that everyone is talking about Europe's demographic death, it is time to point out that there exists a way out: convert European Muslims to Christianity. The reported front-runner at the Vatican conclave that began on Monday, Cardinal Joseph Ratzinger, is one of the few Church leaders unafraid to raise the subject. [1] Hedonistic dissipation well may have condemned the existing Europeans to infecundity and extinction, but that does not prevent Europe from getting new ones. It has been done before....

Now that John Paul II has been buried, Catholic voices are sounding the alarm about the coming Islamicization of Europe. In the future imagined by John Paul II's biographer George Weigel, "The muezzin summons the faithful to prayer from the central loggia of St Peter's in Rome, while Notre-Dame has been transformed into Hagia Sophia on the Seine - a great Christian church become an Islamic museum." [2]

If the Italians and French were all Muslim I suppose the French and the Italians would welcome those conversions.

Precisely how the Church might go about proselytizing Muslims is a different matter, and a dangerous one, considering that Islam decrees the death penalty for apostates...

Yes, there is that. The Saudis whacked a fellow awhile back for possession of a Bible.