To be able to complete the mission you have to bring the relic to the client. This is in a house behind a white door. Close to the market.

Did the whole thing. Took me over two hours to kill everyone, triple Shinen and the like, without any detection or suspision. All this to find that the exit point doesn't work. Had to hard-quit. Sorry but i cannot put you 5 stars for it. Shame, really, because it's quite a nice one...