  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 $G$.
