The freeness conjecture for the parking-function dual Hopf algebra

Let ${\bf PQSym}^*$ denote the dual Hopf algebra of parking functions, realized as a dendriform trialgebra with operations induced from the dendriform trialgebra structure on ${\mathbb K}\langle A\rangle^+$. Freeness conjecture. ${\bf PQSym}^*$ is a free dendriform trialgebra. This conjecture concerns the algebraic structure of the dual parking-function Hopf algebra and is proposed on the basis of numerical evidence; no resolution is stated here.