Flow scalability characterizes weighted α-fairness for arbitrary network topologies

Let a network have routes with utilities $U_r$ and suppose its allocation is flow scalable, meaning that scaling the capacities and offered flows according to the network's flow-scaling transformation preserves the normalized allocation. A utility function is weighted $\alpha$-fair when, for a common $\alpha>0$ and positive route weights $w_r$, it has the form

$$U_r(x_r)= \begin{cases} w_r\dfrac{x_r^{1-\alpha}}{1-\alpha}+c_r & \text{if } \alpha\ne 1,\\ w_r\log x_r+c_r & \text{if } \alpha=1. \end{cases}$$

Flow-scalability conjecture. A network is flow scalable if and only if it maximizes a weighted $\alpha$-fair utility function.

The preceding result establishes this equivalence for networks satisfying the local traffic condition, and any network topology can be made to satisfy that condition by adding a local traffic route to each link. The conjecture asks whether the equivalence holds for every network topology; the supplied text gives no resolution beyond the stated special case.