If you lose the scepter, do you need the key again to get it from the chest?

I got the music key much earlier than 30 minutes before 1pm. I believe it is actually late in the draft pool, not time. Possibly the last 5-10 rooms.

@jasonrodriguez: Yes I was on a rather long run and waited over an hour. 1:30 pm worked. Thanks for putting all this together!

didnt work for me. Perhaps you have to be facing north for it to happen? or maybe it only happens at 1:30 pm not am.