Based on the Lost pages event and the Halloween Tower quests, no it won't stay in your log until next time. One nice thing is the B.O.X. quest does not require you to start on any particular box. If say, the last box is the only one you didn't get to, you ought to be able to just do that one if the quest returns. There does seem to be a badge for finishing all the boxes, so you might need to run them all again in one event to get that badge. I'm just guessing, it might be that if you have the earlier box badges, and finish the ones you lacked, you might still get the Time Weaver badge too, without running all of them. That remains to be tested.