Heavy-traffic queue-length distribution conjecture for roving server networks

Let $L_i$ be the queue length at queue $i$, let $\mathcal{L}^{\textit{fluid}}_{i,k}$ be the standardised number of type-$i$ particles in the fluid model during visit period $V_k$, and let $\Gamma$ be independent of $\mathcal{L}^{\textit{fluid}}_{i,k}$. Let $\alpha$, $\delta$, and $\mu$ be as defined above. Queue-length conjecture. As $\rho\uparrow1$, the scaled queue length $(1-\rho)L_i$ converges in distribution to the product of two independent random variables, with

$$(1-\rho)L_i\xrightarrow{d}\Gamma\times\mathcal{L}^{\textit{fluid}}_{i,k}\qquad\text{with probability }\hat\rho_k,$$

for $i=1,\dots,N$ and $k=i,\dots,i+N-1$. Here $\Gamma$ has a Gamma distribution with parameters $\alpha+1$ and $\delta\mu$. The source notes that a theorem and proof are available under Poisson arrivals; the general arrival setting remains conjectural.