Fine's existence conjecture for the extended g-vector

Let $X$ be a convex polytope, and let the extended $g$-vector be the vector whose components are the linear functions on flag vectors specified by the axioms in the previous section. Fine's existence conjecture. There is a $g$-vector that satisfies the stated axioms. This is the first of two conjectures intended to determine the extended $g$-vector in each dimension; together with the finite-calculation conjecture, it would yield linear inequalities on convex-polytope flag vectors.