ping
I’d be very surprised: 1) if the Vatican accepted overseeing the Temple Mount, and 2) If the Vatican accepted such overseeing, that the muslim organization (Wafq?) that currently holds the keys, so to speak (Dayan let them have it back after the ‘67 war) would even permit any Christian church/organization to have any control.
I don’t know of any instance where islam every agreed to any non-muslim entity controlling any place where a muslim’s foot stood for 30 seconds without a war happening first. If Pope Benedict XVI doesn’t tell Peres “thanks, but no thanks”, the Wafq will kybash it. The muslims think the Temple Mount is theirs, and their “excavations” where they dump Jewish artifacts in the garbage to try and eliminate any traces of pre-muslim occupation of the Mount show that they would not plan on giving control of that area to anyone without a major rhubarb.