  The notion of a holographic entropy cone has recently been introduced and it
has been proven that this cone is polyhedral. However, the original definition
was fully geometric and did not strictly require a holographic duality. We
introduce a new definition of the cone, insisting that the geometries used for
its construction should be dual to states of a CFT. As a result, the
polyhedrality of this holographic cone does not immediately follow. A numerical
evaluation of the Euclidean action for the geometries that realize extremal
rays of the original cone indicates that these are subdominant bulk phases of
natural path integrals. The result challenges the expectation that such
geometries are in fact dual to CFT states.
