Torus Queen Independence

Define a queen on ${\mathbb{Z}}_n^d$ with admissible moves parallel to $\mathbf{x}\in \{-1,0,1{\}}^d$ at arbitrary length. How many queens can be placed on ${\mathbb{Z}}_n^d$ without any two in conflict? In two dimensions, this problem was initiated by Pólya in 1918 and resolved by Monsky in 1989. We give the first known results in $d$ dimensions, showing that the trivial upper bound $n^{d-1}$ cannot be attained if $n$ is a multiple of $5$, not $25$. We demonstrate, for every $d$, how $n^{d-1}-O(n^{d-2})$ queens can be placed independently.