The asymptotic Faber–Krahn shape conjecture for subgraphs of the integer lattice

Let $G_n$ be any sequence of subgraphs in $Z^d$ such that $|G_n|=n$ and $\lambda_D(G_n)=\lambda_D^{(n)}$. Let $D\subset R^d$ denote the unit ball, and let $G_n^*$ be the rescaled realization of $G_n$. The Hausdorff distance between $G_n^*$ and $D$ is measured after translating the graphs.

Asymptotic Faber–Krahn shape conjecture. After possibly translating the $G_n$, the Hausdorff distance between $G_n^*$ and $D$ converges to $0$ as $n\to\infty$.

This conjecture predicts that discrete domains minimizing the Dirichlet eigenvalue asymptotically become spherical, extending the continuum Faber–Krahn principle from $R^d$ to the integer lattice. The supplied text does not state whether the claim has been resolved.