  The domain of allowed von Neumann entropies of a holographic field theory
carves out a polyhedral cone -- the holographic entropy cone -- in entropy
space. Such polyhedral cones are characterized by their extreme rays. For an
arbitrary number of parties, it is known that the so-called perfect tensors are
extreme rays. In this work, we constrain the form of the remaining extreme rays
by showing that they correspond to geometries with vanishing mutual information
between any two parties, ensuring the absence of Bell pair type entanglement
between them. This is tantamount to proving that besides subadditivity, all
non-redundant holographic entropy inequalities are superbalanced, i.e. not only
do UV divergences cancel in the inequality itself (assuming smooth entangling
surfaces), but also in the purification thereof.
