This happened to me back when the game first came out and tired to contact bungie about this issue before but nothing ever came of it.
It was in which I was completing the rat king quest steps and the quest bugged out on one of the steps which caused to to stop getting quest progress, to which I though I’d just delete the quest and start again which would be a big deal, but I was obviously wrong as I still don’t have rat king to this day, is there any way for me to bungie to fix this problem in either just giving me the quest back or giving me the weapon?
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As @Luca270999 said, the Rat King quest is a bit unusual in that it will drop for others in the fireteam who don't have the Rat King when one other teammate has it equipped. I've found the best way to have it drop is to run a nightfall with the Rat King equipped. At the end of the Nightfall, the quest dropped for the teammate I was helping.