Flow-capacity conjecture for directed hypercube separation distance

Let $S$ be a source set in the directed hypercube with separation distance $r$. Consider a flow network with unit edge capacities and vertex capacities $r^2$. Flow-capacity conjecture. The maxflow is at least

$$\Omega\!\left(r\,\mu^+(S)\right).$$

This would relate edge-capacitated and vertex-capacitated flows in the directed hypercube and is motivated by seeking a flow-based proof of the KMS theorem; the source gives no resolution of the conjecture.