Honestly I have the same question too . So, after Red War we moved to this new place to start everything. The doors are already there, locked and no one know whats behind until now. The Vanguards and Guardians have occupied a place where they don't have much idea. Our "friend" Spider knows it for...I believe it has longer than we could believe . A guy, named himself Drifter is there hosting the gambit thing who we have very limited information about him. In that case, I would say, No one in this Tower can re-gain my trust.
Well umm. The drifter can give you a Trust if ya play enough gambit..