Free dendriform trialgebra conjecture for colored parking-function quasi-symmetric functions

Let $l$ be a positive integer, and let ${\bf PQSym}^{(l)}$ denote the colored parking-function quasi-symmetric-function algebra introduced in the paper. A dendriform trialgebra is an algebra equipped with three compatible dendriform operations.

Free dendriform trialgebra conjecture. The algebra ${\bf PQSym}^{(l)}$ is a free dendriform trialgebra.

This extends the corresponding conjecture for ${\bf PQSym}$. The statement concerns the tridendriform structure suggested by the preceding bidendriform structure; its resolution is not given in the supplied text.