Coarse Gallai conjecture for A-paths

Let $G$ be a graph and let $A$ be a subset of the vertices of $G$. An $A$-path is a path in $G$ whose two endpoints belong to $A$. For subsets $X$ of vertices of $G$ and integers $r\geq 0$, let $B_G(X,r)$ denote the set of vertices at distance at most $r$ from $X$. Two subgraphs are at distance at least $d$ if the distance between their vertex sets is at least $d$. Coarse Gallai conjecture. There exist functions $f:\mathbb N\to\mathbb N$ and $g:\mathbb N\to\mathbb N$ such that for all positive integers $k$ and $d$, for every graph $G$ and every subset $A$ of the vertices of $G$, either $G$ contains $k$ $A$-paths which are pairwise at distance at least $d$, or there exists a set $X$ of the vertices of $G$ with $|X|\leq f(k)$ such that every $A$-path in $G$ contains a vertex of $B_G(X,g(d))$. This strengthens the coarse Erdős–Pósa-type statement by requiring the radius of the hitting balls to depend only on $d$, while the size of the hitting set depends only on $k$.