There is no reason to believe that the Palestinians would respect Jewish and Christian holy places. If you remember what happened when Jordan had control of East Jerusalem, you would know this. What the Palestinians recently did in Nablus is also in violation of the Oslo Accords. If they can't respect Jewish holy places now, why would they when they have complete control?

The person in charge of Muslim holy places in Jerusalem, the Mufti of Jerusalem, made a very ignorant statement,

There is not the smallest indication of the existence of a Jewish temple on this place in the past. In the whole city, there is not even a single stone indicating Jewish history. (from the German Die Welt)

This is not only contradicted by facts and history, but also by his very own Qur'an, (Suras 17 and 34). Regardless of any stupidity in taking his position, he would be the one in charge of holy sites if there were to be Palestinian Authority control over Jerusalem.