The saturated-flow formula for the PreLie quotient series

Let $\mathscr{Y}=\mathscr{E}\circ\mathscr{D}^{-1}$ be the quotient series in the group $\mathsf{G}_{\operatorname{PreLie}}$, let $\mathscr{Y}_T$ be the coefficient of a rooted tree $T$ in $\mathscr{Y}$, let $\mathscr{E}_T^s$ be the generating series for saturated flows on $T$, and let $L(T)$ be the number of leaves of $T$. Saturated-flow quotient formula. The coefficient $\mathscr{Y}_T$ is the monomial

$$\mathscr{Y}_T=(-1)^{L(T)-1}\mathscr{E}_T^s.$$

This formula would give a simple description of the quotient series in terms of saturated flows and the leaf count. The source provides no resolution status.