A stability conjecture on bandwidth sharing networks

We consider a queueing model where documents are simultaneously transferred over a communication network. The bandwidth allocated to each document transfer is assumed to be the solution of a utility optimization problem. Under a natural stability condition and under the assumption that document arrivals are Poisson and that document sizes are independent exponential distributions, such queueing models have been proven to be positive recurrent. It has been conjectured for a decade that the assumption of exponentially distributed documents can be removed. There exist numerous generalizations without this exponential assumption, but a general proof remains elusive.