The limiting generating-function conjecture for slowly controlled Markovian queues

Let $Q$ be the queue length and $S$ the server speed, and let $u$ be the measurement-rate parameter tending to zero. Define

$$\hat{P}(x,y):=\lim_{\nu\to 0}P(x^\nu,y^\nu),$$

the generating function of the scaled pair $(\nu Q,\nu S)$ in the limit $\nu\to 0$. Limiting generating-function conjecture. The limiting generating function is

$$\hat{P}(x,y)=\frac{1}{2}\frac{1}{1-\lambda\log(x)}+\frac{1}{2}\frac{1}{1-\lambda\log(y)}.$$

This conjecture formalizes the proposed cyclic fluid-scale behavior when measurements occur at a much slower rate: the queue alternates between growth while the server is stopped and rapid drainage after a control instant. The authors state that they were unable to formalize this intuition; no resolution is given here.