Kalaiʹs squeezed 3-spheres are polytopal

In 1988, KALAI [4] extended a construction of BlLLERA and LEE to produce many triangulated (d — 1)-spheres. In fact, in view of upper bounds on the number of simplicial d-polytopes by GOODMAN and POLLACK [2, 3], he derived that for every dimension d > 5, most of these (d−1)-spheres are not polytopal. However, for d = 4, this reasoning fails. We can now show that, as already conjectured by KALAI, all of his 3-spheres are in fact polytopal.