The bounded-degree induced Menger conjecture

Let $G$ be a graph, let $\Delta(G)$ denote its maximum degree, and let $X,Y\subseteq V(G)$. A set of vertices separates $X$ and $Y$ if it intersects every $X$-$Y$-path; paths are pairwise at distance at least $d$ when their graph distance in $G$ is at least $d$. The bounded-degree induced Menger conjecture. For every $d,\Delta\in\mathbb{N}$, there exists a constant $C=C(d,\Delta)>0$ such that, for every $k\in\mathbb{N}$ and every graph $G$ with $\Delta(G)\leq\Delta$, together with $X,Y\subseteq V(G)$, there exists either

$$k\text{ }X\text{-}Y\text{-paths pairwise at distance at least }d\text{ in }G,$$

or a set of fewer than $Ck$ vertices in $G$ that separates $X$ and $Y$. This is presented as a necessary consequence of the stronger coarse Menger conjecture of Albrechtsen et al. The source does not report a resolution of this bounded-degree formulation.