This can't be true.
Matthew 24:1-2, "And Jesus went out, and departed from the temple: and his disciples came to him for to shew him the buildings of the temple. And Jesus said unto them, See ye not all these things? verily I say unto you, There shall not be left here one stone upon another, that shall not be thrown down."
The Temple was completely destroyed, and according to Josephus, the entire area where the Temple stood was so thoroughly destroyed that anyone standing upon the ground where it stood would never know it. Plus, the Jewish people were away from Jerusalem for so long they actually forgot where the Temple stood.
What they now claim as the foundation of the Temple and the Western Wall is nothing more than the foundation of a Roman fortification.
Also,from Flavius Josephus, "War of the Jews, Book 7, Chapter 1, Paragraph 1, "NOW as soon as the army had no more people to slay or to plunder, because there remained none to be the objects of their fury, (for they would not have spared any, had there remained any other work to be done,) Caesar gave orders that they should now demolish the entire city and temple, but should leave as many of the towers standing as were of the greatest eminency; that is, Phasaelus, and Hippicus, and Mariamne; and so much of the wall as enclosed the city on the west side. This wall was spared, in order to afford a camp for such as were to lie in garrison, as were the towers also spared, in order to demonstrate to posterity what kind of city it was, and how well fortified, which the Roman valor had subdued; but for all the rest of the wall, it was so thoroughly laid even with the ground by those that dug it up to the foundation, that there was left nothing to make those that came thither believe it had ever been inhabited. This was the end which Jerusalem came to by the madness of those that were for innovations; a city otherwise of great magnificence, and of mighty fame among all mankind."
The entire city and the Temple, except for a wall and a few towers, was completely leveled, just as Jesus said it would happen. So there is no way the so-called "Temple Mount" can be place where Solomon's and Herod's Temple stood.
That was the second temple period coming to an end, Herod had built quite a complex, but Solomon's temple was indeed on the Rock. During the Roman War (or Jewish War, as the turncoat Josephus called it) various factions held various parts of that area. When the temple enclosure fell, in an artillery assault (not gunpowder, Roman artillery), the projectiles came in with such intensity the walls couldn't be manned, and ultimately those under siege who could still make it fled into the temple. Obviously it didn't end well. The late Michael Grant, classical historian, assembled quite a thorough account of the period in his "Jews in the Roman World". Ultimately up to 15 percent of the Empire was Jewish.