Linear Erdős–Pósa bound for models of forests

Let $F$ be a forest and let $R \subseteq V(F)$. For a graph $G$, a set $S \subseteq V(G)$, and a positive integer $k$, consider models of $(F,R)$ in $(G,S)$.

Linear Erdős–Pósa conjecture. At least one of the following holds:

1. There are $k$ pairwise vertex-disjoint models of $(F,R)$ in $(G,S)$.
2. There is a set $X \subseteq V(G)$ with

$$|X| \leq |V(F)|(k-1)$$

such that there is no model of $(F,R)$ in $(G-X,S \setminus X)$.

This would give a bound linear in the size of the forest, improving the dependence on $|V(T)|$ in the established Erdős–Pósa result for $S$-rooted models of a fixed tree. The conjecture is stated in the more general setting of $(F,R)$-models and is open in the supplied source.