Another example of eight neighborly tetrahedra was given by S. Wilson and the author ;
here, too, there are two quadruples of tetrahedra separated by a plane which contains a facet of each
one of the tetrahedra; this example is shown in Figure 2.
Figure 2. The bases of eight tetrahedra, in another example
of a neighborly family, taken from .
Baston stated an argument which implies that a neighborly family of tetrahedra contains at
most seventeen members. He conjectured that the maximum number of tetrahedra in a neighborly
family is eight. Baston  showed in 1965 that this maximum is at most nine, and he repeated
Bagemihl's conjecture that the maximum is eight.
Bagemihl's conjecture had been repeatedly mentioned in the literature: by Danzer, Grunbaum
and Klee  in 1963, by Grunbaum  in 1967 and in particular by Klee  in 1969 (it is
also mentioned by Klee in his research-film called "Shapes of the Future", part two, produced by
the M.A.A. at about 1972); it is also mentioned by Perles , by kassem  and by us [17,18].