The half-integral coarse Menger conjecture

Let $G$ be a graph and let $X,Y\subseteq V(G)$. An $X$-$Y$ path is a path with at least one end in $X$ and at least one end in $Y$. A set is $(a,b)$-centered if it is contained in the union of at most $a$ balls of radius at most $b$. The half-integral coarse Menger conjecture. There exist functions $f,g:\mathbb N\times\mathbb R_{>0}\rightarrow\mathbb R$ such that for any $k\in\mathbb N$, $r\in\mathbb R_{>0}$, graph $G$ and subsets $X,Y$ of $V(G)$, either there exist $k$ $X$-$Y$ paths $P_1,P_2,\ldots,P_k$ in $G$ such that every ball of radius $r$ in $G$ intersects at most two of $P_1,P_2,\ldots,P_k$, or there exists an $(f(k,r),g(k,r))$-centered set intersecting all $X$-$Y$ paths in $G$. This is presented as the half-integral weakening of the weak coarse Menger conjecture. The source gives no resolution.