The Induced Menger Conjecture

Let $G$ be a graph and let $X,Y\subseteq V(G)$. An $(X,Y)$-path is a path with one endpoint in $X$ and the other in $Y$, and two paths are anti-complete if they are vertex-disjoint and no edge of $G$ has one endpoint in each path. Write $N_G[Z]$ for the closed neighbourhood of $Z$. The Induced Menger Conjecture. For every $k\in\mathbb{N}$, every graph $G$ and $X,Y\subseteq V(G)$ satisfies one of the following: $G$ contains $k$ pairwise anti-complete $(X,Y)$-paths; or there exists a set $Z\subseteq V(G)$ with $|Z|\leqslant k-1$ such that $G-N_G[Z]$ has no $(X,Y)$-path. This is the induced analogue of Menger's theorem and corresponds to the $d=2$ case of the coarse Menger conjecture; the paper's induced $\mathcal{A}$-path theorem provides evidence but does not resolve it.