Conjecture on the order-complex structure of pure diagrams

Let $\varPi^M_m$ be the finite poset of pure diagrams between $\pi(m)$ and $\pi(M)$, and let $\Delta(\varPi^M_m)$ be its order complex, whose faces are chains in $\varPi^M_m$. Regard this complex geometrically in the vector space $V^M_m$ by sending each poset element to its pure diagram. Order-complex conjecture. Any non-negative linear combination of pure diagrams is a non-negative linear combination of pure diagrams from one chain; equivalently, the convex hull of $\varPi^M_m$ equals the geometric realization of $\Delta(\varPi^M_m)$. This conjecture asserts that the convex hull is assembled from simplices indexed by chains in the pure-diagram poset. The supplied text gives no resolution status.