The half-Eulerian inequality-generation conjecture

A half-Eulerian poset is a graded poset satisfying the half-Eulerian relations, and its flag vector has coordinates $f_S$. A linear form in the flag numbers is an expression

$$\sum_{S\subseteq[1,n]} a_S f_S.$$

Half-Eulerian inequality-generation conjecture. Every linear form that is nonnegative on the flag vectors of all half-Eulerian posets is the sum of a linear form that is nonnegative on the flag vectors of all graded posets and a linear form that vanishes on the flag vectors of all half-Eulerian posets.

This is the precise linear-algebraic formulation of the preceding claim that all inequalities valid for half-Eulerian posets come from inequalities valid for all graded posets. The conjecture concerns linear forms in ${\bf R}^{2^n}$, although half-Eulerian flag vectors span only a subspace; its resolution would describe the half-Eulerian inequalities modulo the linear relations vanishing on that subspace.