Non-negativity conjecture for the coefficients C

Let $\phi$, $\varphi$, and $\psi$ be the set compositions used in the paper, and let $C_{\varphi}^{phipsi}$ denote the corresponding coefficients in the peak-algebra computation.

Non-negativity conjecture. The coefficients

$$C_{\varphi}^{\phi\psi}$$

are non-negative.

The surrounding discussion says that this follows in the special case $\psi=\phi$ if the preceding Eulerian–Catalan identity holds, while a cancellation-free formula remains open.