I had this problem today as well. I joined a friend nearly done with the mission and didn't get credit. I did the mission by myself after and still didn't get credit.
I Then abandoned the lament quest and reacquired it from the tower kiosk and tried the mission again, this time it progressed.
It does seem bugged but if you abandon the quest you can get it back from the step you left off. It may take a couple tries hopefully just one after getting the quest again to progress it.
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Yay! After a week of frustration, this step finally completed. I abandonded the quest, picked it up again from the tower, immediately ran the exo challenge, and now I'm on to the next step: killing vex with finishers. Thanks for the fix!