[math-fun] Find a polyhedron with all vertices on its circumsphere, but without all faces touching its insphere

There is one with 6 faces, anyhow. (Far below if you want to try to find one without spoilage.) —Dan Let T be a skinny isosceles triangle with apex A. Let the polyhedron P be the result of truncating [the product T x [0, 1] of T with the unit interval] by cutting off a neighborhood N of {A} x [0, 1], thereby creating a new face — a skinny rectangle R. Then P has all 8 of its vertices on its circumsphere C. But clearly C would not touch the rectangle R if N is small enough. —Dan