Density of large holes among power-free lattice points

For fixed integers $d,r\ge1$ with $dr\ge2$, the $r$-free points of $\mathbb{Z}^d$ are the vectors whose coordinate gcd is not divisible by the $r$th power of any prime. We consider the set $W_{d,r}$ of non-$r$-free points in $\mathbb{Z}^d$ as the vertex set of a graph with nearest-neighbour adjacenc