Final Fantasy XIV Database - Ffxivpro.com

I'm guessing the reason it depops and respawns after getting an illusion is so people in other locations of the zone have a chance at it rather than everyone hording it in one specific location. Also, it's good that it repops quickly and still keeps it's time restriction for those who recently opened it for treasure so that people who actually need it (AF, maps) can get it and get out. Everyone else has to wait their turn since treasure is minimal priority.