So, I did the timed quest... that didn't clear it either. I looked further into it. It was apparently exactly the "suppress" quest. Thanks for your help.
Converse with fellow players
@KaoriRevan Very odd.
Also, ARKS Quests are always available, not just when they are the featured quest.
I'm still very new to how everything works. I'm currently in Daybreak providence as I've been typing these. So far no spawns for my 4th attempt. I've no idea how to access a supress only quest unless it is featured.
I can't check the CO right now, but it could be a level requirement thing (usually noted at the bottom of the CO description).
Also, you can find it as a guaranteed spawn in the ARKS Quest Suppress Anjadeurili and he Time Attack Quest, Rush Practice: Harukotan.
I would have done the supression quest if it was currently up. I didn't consider that the timed quest would count... and it says Normal and Up... I'm doing this on the highest teir there is for this area.
The quest targets Anjadeurili, which has the lowest spawn rate for mini bosses in Daybreak and is only found there and I can't get the quest to complete. I have beaten no less than THREE of them, even took out the level's final boss and still it didn't complete the quest.