Face-avoidance conjecture for FM-tetrahedra

An FM-tetrahedron is a tetrahedron formed by four pairwise disjoint spheres, with a support sphere tangent to all four and satisfying the FM-tetrahedron conditions described in the paper. For each sphere, the face determined by the centers of the other three spheres is the corresponding triangular face in the tetrahedron of centers. Face-avoidance conjecture. In an FM-tetrahedron, no sphere intersects the face defined by the centers of the three other spheres. The conjecture proposes a three-dimensional analogue of the second equivalent characterization of FM-triangles; the supplied text does not indicate whether it has been proved or disproved.