Uniform limiting distribution in the profit-driven regime

Let $\\{\epsilon_\eta\\}_{\eta>0}$ and $\\{\tau_\eta\\}_{\eta>0}$ satisfy

$$\lim_{\eta\rightarrow\infty}\epsilon_\eta\tau_\eta=\infty.$$

For each $\eta>0$, consider the positive recurrent DTMC $\\{z_\eta(k):k\in\mathbb{Z}_+\\}$ under functions $\phi^c(\cdot)$ and $\phi^s(\cdot)$ satisfying the negative-drift condition, and let $\bar{z}_\eta$ denote its steady-state random variable. Let $\Phi^\star$ be the nonempty set of minimizers of

$$\int_0^x\bigl(\phi^s(t)-\phi^c(t)\bigr)\\,dt.$$

Profit-driven regime conjecture. The scaled steady-state variable should satisfy

$$\frac{\bar{z}_\eta}{\tau_\eta}\overset{D}{\longrightarrow}\mathcal{U}(\Phi^\star).$$

The conjecture predicts that, as the product $\epsilon_\eta\tau_\eta$ diverges, the limiting distribution is uniform on the minimizers of the potential function. The paper notes that proving this is difficult; the corresponding uniform limit is proved only in a special case.