Refined recurrence conjecture for rooted trees with a specified lower critical node

Let $\mathcal{R}_{n,k}$ be the class of rooted trees under consideration, and let $\lambda(T)$ denote the lower critical node of a tree $T$; the notation $\mathcal{R}_{n,k}[\lambda=i]$ denotes the trees in $\mathcal{R}_{n,k}$ whose lower critical node is $i$. For notational simplicity, the condition that the maximum node has positive degree is omitted when $\lambda=i$ is specified.

Refined recurrence conjecture. For $n\geq 3$ and $1\leq i\leq n-2$, we have

$$|\mathcal{R}_{n,k}[\lambda=i]|=(n-2)|\mathcal{R}_{n-1,k}[\lambda=i]|+(n+k-3)|\mathcal{R}_{n-1,k-1}[\lambda=i]|.$$

This conjecture proposes a recurrence refining the recurrence for the numbers $f_{n,k}=|\mathcal{R}_{n,k}|$ by tracking the lower critical node. The paper presents it as an open problem arising from the bijections developed there; no proof or resolution is given.