Consider a triangulated ball $D$, and assume that $\omega$ is an assignment of real weights to the simplices of $D$, including the empty one, such that for every maximal simplex $F$ and every simplex $G$ in $F$, we have

$\sum_{x \in [G,F]} \omega(x) \ge 0.$

Is it true that

$\sum_{x \in D} \omega(x) \ge 0$?

Here $[G,F]$ is the interval of $G$ to $F$, that is, all simplices containing $G$ and contained in $F$.

Edit: for Cohen-Macaulay complexes (which are more general than balls) the answer is no. Raman Sanyal showed that the dual linear program is infeasible for an example of nonpartitionable complexes. I will leave it open since my original question is unsolved, but that provides an indication.