Higher-dimensional affine overlap conjecture

Let $G$ be the group of non-degenerate, equivalently bijective, affine mappings of $\mathbb R^d$. Let $f_G^d(N,k)$ denote the corresponding maximum number of maps having at least $k$ common points with an $N$-point set. The higher-dimensional affine overlap conjecture.

$$f_G^d(N,k)=O\left(\frac{N^{2d+2}}{k^{2d+1}}\right).$$

The conjecture extends the planar affine bound. The source says that the case $d=2$ is its main theorem and that the result can also be extended to $d=3$, while nothing is known there for $d\ge4$.