Discrete positive mass conjecture for asymptotically flat graphs

Let $G=(V,E,w_G)$ be an asymptotically flat graph: outside a finite set, it is weighted isomorphic to the complement of a finite cube in a weighted grid graph $(\mathbb{Z}^n,w)$, where $p>n\geq 2$, $w(x,y)=1+o(1)$ for adjacent vertices, $\operatorname{Abs}(x)=O(|x|^{-p})$, and $|R(x)|=O(|x|^{-p})$. Its scalar curvature is $R(x)=\sum_{y\in V:y\sim x}\kappa(x,y)$, and its discrete ADM energy is

$$E(G)=\frac{1}{2^n n}\lim_{r\rightarrow\infty}\left(\sum_{e\in E_r}w(e)-\sum_{\tau\in \widetilde{E}_r}w(\tau)\right).$$

Discrete positive mass conjecture. For an asymptotically flat graph $G$ with non-negative scalar curvature, one has $E(G)\geq 0$. Moreover, $E(G)=0$ if and only if $G$ is a standard grid graph.

This is a graph-theoretic discrete analogue of the positive mass theorem for asymptotically flat manifolds. The conjecture asserts both non-negativity of the discrete ADM energy and rigidity in the zero-energy case; the supplied source does not indicate whether it has been resolved.