Embedding Borel graphs into grids of asymptotically optimal dimension

Let G be a Borel graph all of whose finite subgraphs embed into the d-dimensional grid with diagonals. We show that then G itself admits a Borel embedding into the Schreier graph of a free Borel action of ZO(d). This strengthens an earlier result of the authors, in which $O(d)$ is replaced by O(ρ log ρ), where ρ is the polynomial growth rate of G.