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 ${\mathbb{Z}}^{O(d)}$. This strengthens an earlier result of the authors, in which $O(d)$ is replaced by $O(\rho \log \rho )$, where $\rho$ is the polynomial growth rate of $G$.