But that doesn't set a very good precendent, we might as well ask half the countries of the world to hand over their cultural treasures as they obviously can't (or won't) take care of them like we would.
It would be very embarrassing, too, given how much of a luxury problem cultural heritage is.
And what if they do? If the Egyptian government decides it's a good idea to dismantle the pyramids what are we going to do about it? They may 'belong to all mankind', but all mankind doesn't own them.
And in the West we're not exactly innocent when it comes to purposely destroying ancient treasures. Even today, the way we're treating places like Babylon doesn't inspire confidence.
As for the Elgin marbles, there doesn't seem to be a happy solution. Perhaps some sort of joint ownership and regular loans would be a sensible approach, especially considering how grand travelling exhibitions seem to have been doing wonders for museums.
On the other hand, what could happen when they first touch Greek soil might be all too predictable.