The campaign must be completed on each character. You have your progress set to private so we cannot check anything.
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Thank you for your reply, I checked my privacy settings, and it seems that everything is set to "Show", only my vault items were not shown, I have adjusted that to show now. I understand that the story quest is not shared by different characters. My situation is that I did a bit of the Witch Queen story quests solo on legendary difficulty, and then started it from scratch on a new character (my Warlock) with my friend. He was the fireteam leader who started the quest, but afterward, we noticed this bugged Arrival mission quest on both our characters. He actually did the story quests on just one of his characters, so I do not believe the issue is linked to that. Let me know if I can do anything else regarding the privacy settings or share any other information so that helping me solve this would be easier.