I can confirm I'm in the same boat. I've completed the new light main missions (got the 3 class specific emblems), the tower intro quest x3 and I just finished the gambit step on my hunter after I had abandoned it earlier due to the -999,999/4500 bug when I previously reset my infamy. Nothing left to pick up in the kiosk.

Thanks for getting back to me quickly. Yes I have completed everything that the quest had to offer. there isnt anything in my kiosk or in my director leading me to continue the quest I also checked DIM to make sure I didn't have any active quests that I was to blind to see in game. Nothing in there either I didn't see any triumphs related to this. I definitely completed the new new light quest that has to do with Shaw Han and the tower intro quests on my Warlock. the only thing I can think of that may have messed it up is that I played through some of the missions with a friend that had gotten the game recently. I did this after I had abandoned the gambit quest step and it was in my kiosk and before I completed it earlier this morning. I don't know if that messed with my full completion of the new light missions and the tower intro. Just a thought