Nonexistence of face-to-face four-dimensional tilings in the Lee metric

A family of nn-dimensional Lee spheres LL is a tiling   of RnRn, if ∪L=Rn∪L=Rn and for every Lu,Lv∈LLu,Lv∈L, the intersection Lu∩LvLu∩Lv is contained in the boundary of LuLu. If neighboring Lee spheres meet along entire (n−1)(n−1)-dimensional faces, then LL is called a face-to-face tiling  . We prove the nonexistence of a face-to-face tiling of R4R4 with Lee spheres of different radii.