We investigate many-body spin squeezing dynamics in an XXZ model with interactions that fall off with distance $r$ as $1/r^alpha$ in $D=2$ and $3$ spatial dimensions. In stark contrast to the Ising model, we find a broad parameter regime where spin squeezing comparable to the infinite-range $alpha=0$ limit is achievable even when interactions are short-ranged, $alpha>D$. A region of collective behavior in which optimal squeezing grows with system size extends all the way to the $alphatoinfty$ limit of nearest-neighbor interactions. Our predictions, made using the discrete truncated Wigner approximation (DTWA), are testable in a variety of experimental cold atomic, molecular, and optical platforms.
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