I'm not saying this is the best method, but if you happen to have two hunters in your fireteam with invisibility buffed (including 'Don't touch me' gauntlets), this may save you some time at the beginning of your raid.
[b]Method[/b]
The first hunter begins to run alone through the labyrinth. Once the first lantern explodes the second hunter sets off and should hopefully not encounter many enemies. The first hunter continues to the end to activate the bridge formation. By the time the second hunter reaches the bridge 'The path forward is clear' and they can activate their super (Bladedancer with Vanish) to complete the mission.
The other fireteam members can wait an the beginning or be invited before the mission is completed to gain the rewards.
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You can solo it with any class