Linear routing strengthening of the Langberg–Médard conjecture

Let a $k$-pair network have sources $s_1,\dots,s_k$ and sinks $t_1,\dots,t_k$, and suppose it is strongly reachable, meaning that for every sink $t_j$ there are edge-disjoint directed paths from all sources to $t_j$. A routing solution is linear if each commodity flow is a linear combination of the path flows in the network. Linear-routing strengthening of the Langberg–Médard conjecture. Each strongly reachable $k$-pair network has a linear routing solution. The claim strengthens the Langberg–Médard conjecture by requiring a linear rather than arbitrary routing solution. No resolution is given in the supplied text.