Apparently, the buttons being all over the place is the answer to this one. My god this event is outright AWFUL, why are there loading screens when you go in and out of combat? SERIOUSLY.For this event, it looks like ~140 if you go through in one go. Notice that "one go" doesn't mean in one sitting. However, you cannot host anything until you are finished.
So, this event in a nutshell:
1. Do your dailies, GO and AQ.
2. Start your run. For veterans, simply take Solomon with Desertion EX skill - the highest HP mob is 1m, so Solomon can easily solo EVERYTHING. Bring 6x Himes that you want to level if you have any. Newer players might want Andromeda(/Cass) since HP carries over from battle to battle, but skill CDs are reset.
3. Once you've done ~140 floors ( = ~2100 AP = ~35 HEs for rank149 people), you will have everything.
Future Labyrinths will be bringing "game over" bosses which force you to restart your run. This isn't the case in this one, so you should do the entire thing in one go.