Non-negativity conjecture for the cd-index of balanced labeled acyclic digraphs

Let $G$ be a bounded labeled acyclic digraph with a balanced linear edge labeling. Non-negativity conjecture. The $\mathbf{c}\mathbf{d}$-index of $G$ is non-negative. This conjecture extends the non-negativity phenomenon established for the examples constructed in the paper and asks whether balanced linear edge labelings always force coefficientwise non-negativity of the $\mathbf{c}\mathbf{d}$-index.