A Menger-type conjecture for many induced paths

Let $G$ be a graph, let $X,Y\subseteq V(G)$, let $d,k\geq 1$ be integers, and let $\operatorname{dist}_G(P_i,P_j)$ denote the distance between paths in $G$. For a set $Z\subseteq V(G)$, let $B_G(Z,cd)$ denote its radius-$cd$ neighborhood.

Many-path Menger-type conjecture. There exists a constant $c$ such that, for all such $G$, $X$, $Y$, $d$, and $k$, either there exist $k$ disjoint $X-Y$ paths $P_1,\dots,P_k$ satisfying

$$\operatorname{dist}_G(P_i,P_j)\geq d$$

for all distinct $i,j$, or there exists a set $Z\subseteq V(G)$ of size at most $k-1$ such that $B_G(Z,cd)$ intersects every $X-Y$ path.

This conjectures a many-path extension of the paper's two-path theorem, which gives such a constant-radius obstruction for two paths. The corresponding exact characterization is motivated by Menger's theorem but is not known for paths required to be pairwise far apart.