I had the same experience with this quest while I was in a group of 3 people. Not even the person who owned the phase could complete because we had each clicked different boxes. We had to leave phase, break group, reset the quest and do it solo.
Another unfortunate "feature" in the Black Hole Imperial quest Glow was that with the quest auto-complete, you lose the Anti-Rad injector that protects you from the acid water as soon as you click the last panel. Killing mobs no longer drops them, so you have to run across the acid while taking the full damage. They should fix this quest so that the auto-complete shows up when you leave the instance.