So I just finished mine. but I had to re due it do to a simple mess up because of poor wording. You have to kill the special warden and THEN the sunless cell boss. after you kill your warden you have 30 seconds to drop the boss. I was silly and dropped the boss first which de spawned my warden thus forcing me to run it again.
all 3 wardens will be present regardless of which one you need to kill. they show up one at a time at certain health markers of the boss. So all you have to do is kill the other wardens. wound yours and then weaken the boss till hes nearly dead. then you kill your warden and then drop the boss. that's it. It doesn't consume your legendary sword either. so you have 2 now.
Another tip (if you are a night stalker) is the shrieker room can be skipped entirely since the taken leave the door open. just progress about halfway into the room in the middle use vanish in smoke and then run past them through the door. Hope this helps people!
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It glitched for me, I killed the warden then the boss and completed the strike, but it says I never killed Aluk-kul. Anyone else having this problem?