Same issue here. Read some other tips and abandon the quest with all three of my characters, went back to the tower and required the quest, completed another exo challenge and again no avail. I have only tried with my hunter but haven't gotten that far with my other two characters on the Lost Lament.

I have run and completed 2 of the 3 exo challenges and no credit given for step 5 progress. Do all 3 challenges have to be done for it to progress? The way it reads seemed like you only had to do 1 . Today I went ahead and abandoned the quest and reacquired it at the kiosk & will try again. *sigh*

I went to Banshee at the Tower with another character and picked up the quest. I ran the Exo Challenge again and it progressed to the next step. Odd, but workable nonetheless. Thanks for the reply!