The second-moment finiteness conjecture for minimal stationary queue lengths

Let $x_{0;\infty}(0)$ denote the queue length at the origin at time $0$ in the minimal stationary solution of the interference queueing network on the grid, with arrival rate $\lambda$ and interference sequence $(a_j)_{j\in\mathbb{Z}^d}$. Second-moment finiteness conjecture. If

$$\lambda < \frac{1}{\sum_{j \in \mathbb{Z}^d}a_j},$$

then

$$\mathbb{E}[x_{0;\infty}(0)^2] < \infty.$$

The conjecture gives finiteness of the second moment throughout the stability region. The paper states that it was subsequently proved correct using rate-conservation techniques, so the claim is solved.