Stochastic ordering conjecture for needy patients in Erlang-R and closed ward models

Let $Q_1^b(\infty)$, $Q_1^h(\infty)$ and $Q_1^c(\infty)$ denote the stationary numbers of needy patients in the Erlang-R models with blocking and holding and in the closed ward model, respectively. For random variables, write $X\preceq_{\rm st}Y$ when $X$ is stochastically dominated by $Y$. Stochastic ordering conjecture.

$$Q_1^b(\infty)\preceq_{\rm st}Q_1^h(\infty)\preceq_{\rm st}Q_1^c(\infty).$$

The ordering is motivated by simulation results under the scaling considered in the paper and was observed across a multitude of parameter settings for $(\beta,\gamma)$. It compares the stationary needy-patient populations of the blocking, holding and closed ward models; no proof or resolution is supplied here.