The refined flow-series recursion conjecture

Let $T$ be a rooted tree, let $v$ be a vertex of $T$, let $u$ be its parent, let $w$ be a rooted tree grafted at $v$, and let $S$ and $S'$ be the rooted trees defined by removing everything above $v$ and taking the subtree rooted at $v$, respectively. Let $\mathscr{F}_{T,t}$ and $\mathscr{E}_{T,t}$ be rational functions associated with rooted trees, and let $\mathtt{Lnr}(n)$ and $\mathtt{Crl}_n$ denote the linear tree and corolla. Refined flow-series recursion conjecture. There exist rational functions $\mathscr{F}_{T,t}$ such that

$$\mathscr{F}_{T\curvearrowleft_v w,t}=\mathscr{F}_{T\curvearrowleft_u w,t}+(1-t)\mathscr{F}_{S,t}\mathscr{F}_{S',t},$$

and such that

$$\mathscr{F}_{\mathtt{Lnr}(n),t}=\mathscr{E}_{\mathtt{Lnr}(n),t}\qquad\text{for }n\geq1,$$

and

$$\mathscr{F}_{\mathtt{Crl}_n}=\frac{b(-t)^{n-2}}{(1-t)^{n-1}}\qquad\text{for }n\geq2.$$

This conjecture seeks a parameterized refinement of the inductive identity for flow-generating series, with prescribed values on linear trees and corollas. The source provides no resolution status.