With valor, and glory I have had the triumph for win streaks not work. I've gotten 5 in a row on glory, and now have done it on valor. I've tried going to tower, then orbit. Tried changing characters. Nothing. Here's the proof
https://clips.twitch.tv/ColdbloodedTenaciousWallabyAllenHuhu
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You have to earn the unbroken (5 wins) a total of FIVE times in order for that triumph to pop, not just once