I keep strongly suggesting to the government of Israel that it needs to nationalize all holy sites in Israel, because of such foreign claims. If you own your house, you own all of it, no matter if someone claims that they own a bathroom, so can control your use of it.
Once Israel has nationalized all holy sites, it can be very generous about their use via a nominal lease contingent on good behavior. But if a tenant, say of al-Asqa, misbehaves, Israel can revoke their lease and offer it to someone else.
As a real world example, say the tenants are Wahhabi radicals. If they made trouble, the Israelis could kick them out and offer al-Asqa to say, the Sufis.
Not ironically, to strip the Wahhabis of their authority in *Saudi Arabia*, and replace them with Sufis, is a big threat of the Saud family, when the Wahhabis make trouble there. So there is precedent for the Israelis doing this as well.
By Jordan taking this land, they abrogate the treaty. Does this then return the Transjordan lands back under Israel's Control? It would be nice for Israel to get back the 87% of Palestine it gave up for Peace that it never received.