Generalized Lehman–Ron conjecture for directed hypercube layers

Let $L_i$) and $L_j$ be layers of the directed hypercube, with $i<j$ and $r:=j-i$. Let $(S,T;\phi)$ be a matched pair with $S\subseteq L_i$ and $T\subseteq L_j$. Generalized Lehman–Ron conjecture. There are $r$ collections of vertex-disjoint paths between $S$ and $T$ such that their union is edge-disjoint. As the separation between the layers increases, the conjecture predicts proportionally more collections of paths; the paper states that its techniques do not resolve this claim.