Generalized diffusion limit for queue-length processes

Let $U_{n,k}(t)$ denote the scaled fraction of queues with at least $k$ customers, let $F_{n,k}(0)$ denote the corresponding initial tail fractions, and let $u_k(t)$ be the fluid-limit functions. Fix $k\le K$, where $K$ is defined in the paper, and suppose that $d(n)$ is the buffer-size parameter. Assume

$$\lim_{n\to\infty}\frac{n}{d(n)}=\infty.$$

Assume that $U_{n,k}(0)$ is deterministic for all $n$ and $k\le K$, and that there exist $v_1,\dots,v_K\in\mathbb{R}_+$ and $v_1^*,\dots,v_K^*\in\mathbb{R}$ such that

$$\lim_{n\to\infty}\sqrt{\frac{n}{d(n)^k}}\left(U_{n,k}(0)-v_k\right)=v_k^*.$$

Additionally, assume

$$\lim_{n\to\infty}\sqrt{n\,d(n)^{K+1}}\left(F_{n,K+1}(0)+F_{n,K+2}(0)+\cdots\right)=0.$$

Generalized diffusion-limit conjecture. As $n\to\infty$,

$$\sqrt{\frac{n}{d(n)^k}}\left(U_{n,k}(t)-u_k(t)+\frac{1}{d(n)}u_{k+1}(t)\right)\Rightarrow Z_k(t),$$

where $u_{K+1}(t)$ is interpreted as zero. The process $Z_1(t)$ is the solution of

$$Z_1(t)=v_1^*+\sqrt{\lambda}\,B^{(1)}_1(t)-\int_0^t\sqrt{1-e^{-u_1(s)}}\,\mathrm{d}B^{(2)}_1(s)-\int_0^t e^{-u_1(s)}Z_1(s)\,\mathrm{d}s,$$

and, for $k=2,\dots,K$, $Z_k(t)$ is the solution of

$$Z_k(t)=v_k^*+\int_0^t\sqrt{\lambda u_{k-1}(s)}\,\mathrm{d}B^{(1)}_k(s)-\int_0^t\sqrt{u_k(s)}\,\mathrm{d}B^{(2)}_k(s)-\int_0^t Z_k(s)\,\mathrm{d}s,$$

for independent Wiener processes $B^{(1)}_k(t)$ and $B^{(2)}_k(t)$. The conjecture concerns a proposed extension of the paper's proved diffusion limit: the different scaling orders for successive queue-length levels introduce complications for the proof, and the generalization remains unproved in the source.