You might think the answer to that is obvious and well known, but you'd be surprised. First I'm going to quote the account of the Prophecy from Mark 13:1-2 since I think it's the most complete account of exactly what Jesus in this case.
And as he went out of the temple, one of his disciples saith unto him, "Master, see what manner of stones and what buildings are here!" And Jesus answering said unto him, "Seest thou these great buildings? there shall not be left one stone upon another, that shall not be thrown down".
Notice that it isn't JUST about The Temple, it's about all the buildings, plural.
While Matthew and Luke's account of this in their main Olivet Discourse chapters downplay the inclusion of other buildings, Luke 19:44 also refers to not one stone being left, with The Temple not even being the focus, that Prophecy is about the entirety of Jerusalem.
The 9th of Av in AD 70 (presumed to be August 4th on the Roman Calendar) as recorded in Josephus Wars of The Jews Book VI Chapters 4-5 is the day The Temple was destroyed in the sense of not being able to be used as a Temple anymore. Remember what happened to the Notre Dame Cathedral a few years ago? The worst case scenario people were fearing that day is basically what happened to The Temple on the 9th of Av. The next day however as recorded by Josephus in Wars Book VI Chapter 6 there are clearly still standing ruins.
The beginning of Book VII is when Titus demolishes even those ruins and thus this is where most Christians talking about AD 70 via Josephus (both Preterists and Futurists) say the Not One Stone Prophecy was fulfilled. Except Josephus tells us there were three towers that Titus left standing, in my view as long as those three towers were still standing this Prophecy of Jesus was incomplete.
In AD 131 Emperor Hadrian while visiting Jerusalem after ending his extended stay in Egypt announced his plans to rebuilt Jerusalem as a Greco-Roman City with a Temple to Zeus being built over the former site of The Temple. I think the early stages of that project is when even those three remaining towers were torn down.
Then after Hadrian left the Near East for Asia Minor in 132 the Bar Kokchba Revolt broke out. That probably stalled the reconstruction project even though the Rebels never held Jerusalem during that war. Then after the revolt was put down in 135 the project restarted.