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The notion of a holographic entropy cone has recently been introduced and ithas been proven that this cone is polyhedral. However, the original definitionwas fully geometric and did not strictly require a holographic duality. Weintroduce a new definition of the cone, insisting that the geometries used forits construction should be dual to states of a CFT. As a result, thepolyhedrality of this holographic cone does not immediately follow. A numericalevaluation of the Euclidean action for the geometries that realize extremalrays of the original cone indicates that these are subdominant bulk phases ofnatural path integrals. The result challenges the expectation that suchgeometries are in fact dual to CFT states.

Handlebody phases and the polyhedrality of the holographic entropy cone