If I understand correctly, the spin and precession of a top orbiting the black hole will be affected through the Thirring–Lense effect. This is in addition to the static effect of the high gravity. In particular, the black hole will try to drag the top so it spins in the opposite direction to the black hole.
That's been my experience with the black hole down the street.
I still subscribe to the notion that a true top must be supported at a single point and must fall when not spinning. Which implies both a gravitational center and a support. For our tops, that center is Earth's CM. For Sgr A, the Milky Way's central supermassive black hole, it's the
Great Attractor. Now all we need is a volunteer for the support.