The descent-algebra idempotent conjecture for the series Z

Let $\mathsf{Z}$ be the dendriform image of the series $\mathscr{Z}$, let $\mathsf{Z}_n$ be its homogeneous component of degree $n$, and let $b$ be the flow variable. Z-series descent-algebra conjecture. The element $\mathsf{Z}_n$ belongs to the descent algebra and satisfies, in the symmetric group ring of $\mathfrak{S}_n$,

$$\mathsf{Z}_n\cdot\mathsf{Z}_n=nb^{n-1}\mathsf{Z}_n.$$

If the preceding canopy-based coefficient description holds, this would show that the coefficients depend only on the canopy and that the homogeneous components form scaled idempotents in the descent algebra. The source provides no resolution status.