Ended up figuring it out. Nothing happens. I realized that walking up the stairs puts one (approximately) in an inertial reference frame. Surely the stairs moving behind me could not affect the motion of my briefcase.
Haha yeah, I was looking around for something to simulate it but alas I have no such resource at my disposal before this is due. Thanks for your help, I'm sure it will come to me soon
Thanks for your reply. It seems reasonable to me that there is no real force feedback between the gyroscope and the briefcase, so nothing would happen, but the way the question is phrased in the book suggests that something does happen. The question before it asks what would happen if you tried...
1. I am carrying a briefcase in which sits a gyroscope whose large angular velocity points forward horizontally. When I try to climb stairs, what happens to the briefcase?
2. 0=\Delta\vec{}L
3. My intuition says that the briefcase would point downward, but I really don't know how to approach...