Here is my proposal called "Magnetically Suspended Flywheel System", which is closer to the Levitron approach.
The gravitational force is used to suppress "tilt" instability, just like the Post patent. The ring-shaped stator magnet forms inverse heart-shaped magnetic flux lines, and the cone-shaped rotor magnet forms heart-shaped magnetic flux lines. The stable magnetic flux line patterns at the air gap between the ring-shaped stator magnet and the cone-shaped rotor magnet provide axial stability and radial stability. These stabilities are due to a cone-shaped well in a toroidal magnetic field, which is formed by a concentrated first magnetic pole and a distributed second magnetic pole.
Before you dismiss this flywheel system as a crackpot junk, please consider the following, as the late Bill Nye The Science Guy used to say :-).
Rev. Samuel Earnshaw's Theorem assumes Coulomb's law. However, real magnets do not exactly follow the inverse square law (1/r^2), which predicts infinite force between like poles. Instead, like poles of real magnets stick to each other when forced. Here is a text of The Living Atom Theory (Chapter 13) explaining this point, although Uncle Al calls the author an idiot (what else is new?).
Dipoles do not exactly follow the inverse square law (1/r^2) but follow the inverse cube law (1/r^3).
The Lagrange Points between gravitational bodies in space provide stable equilibrium (L4, L5). There are equivalent Magnetic Lagrange Points to be discovered, along with "Sunatori Equilibrium".