Coloring count cones of planar graphs

For a plane near-triangulation G with the outer face bounded by a cycle C, let n(G)* denote the function that to each 4-coloring psi of C assigns the number of ways psi extends to a 4-coloring ofG. The Block-count reducibility argument (which has been developed in connection with attempted proofs of the Four Color Theorem) is equivalent to the statement that the function n(G)* belongs to a certain cone in the space of all functions from 4-colorings of C to real numbers. We investigate the properties of this cone for vertical bar C vertical bar = 5, formulate a conjecture strengthening the Four Color Theorem, and present evidence supporting this conjecture.