Freeness conjecture for the word quasi-symmetric functions trialgebra

Let ${\bf WQSym}$ denote the algebra of word quasi-symmetric functions equipped with its dendriform trialgebra structure, with partial products $\prec$, $\circ$, and $\succ$ satisfying the dendriform trialgebra relations. The sub-trialgebra generated by ${\bf M}_1=\sum_{i\geq 1}a_i$ is the free dendriform trialgebra on one generator.

Freeness conjecture. The dendriform trialgebra ${\bf WQSym}$ is free.

The conjecture proposes that the polynomial realization extends from the known free dendriform trialgebra generated by ${\bf M}_1$ to all of ${\bf WQSym}$. It is based on numerical evidence, and no resolution is given in the source.