Bungie have said once a new quest launches the previous becomes unavailable.
https://www.bungie.net/en/Forums/Post/263889534
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Bit of a bummer as his quests are really cute. Oh well. Maybe you can also help me with this. I actually completed the eight Brave weapon quests but I'm still sitting at 7/8 for "Making the Rounds" triumph where you talk to the different vendors after each one. Am I missing something or is the last one time gated, perhaps? I'm not sure.
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I'm guessing that the last "Round" will probably drop along with Zero Hour for the crafted Outbreak.
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So do you still get Blue Steel for week 2 etc?
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All the triumph says is to find him once, so you should.