Ment
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A polyhedron can lie on its face if its center of gravity is projected onto it. (Here, the center of gravity must be projected perpendicularly onto the plane in which the face it is lying on is located.) Since it turns out that the polyhedron must be convex, otherwise it will not be possible for it to lie on any of its faces, the center of gravity is located inside the polyhedron, which means that if perpendiculars are dropped from it onto all faces, then at least one of them must intersect the face onto which it is dropped.
Guys, this is not a dynamics problem. We are interested in equilibrium states, not how long a particular figure can rotate. An equipotential sphere is in a state of equilibrium by default, albeit an unstable one.