Affine overlap conjecture for proper planar point sets

Let ${\mathcal P}\subset\mathbb R^2$ be any finite point set, let $3\le k\le|{\mathcal P}|$, and let $C>0$. An intersection is proper 2-dimensional up to $C$ in the sense used by the source. The affine overlap conjecture. There is a constant $C^*$, depending on $C$, such that

$$\begin{aligned} \#\{\varphi:\mathbb R^2\to\mathbb R^2\text{ affine}\;;\;&\varphi({\mathcal P})\cap{\mathcal P}\text{ proper 2D up to }C,\\ &|\varphi({\mathcal P})\cap{\mathcal P}|\ge k\}\le C^*\frac{|{\mathcal P}|^6}{k^5}. \end{aligned}$$

This would give the conjectured sharp order of magnitude for affine maps having many common points with a finite planar set; the source says the order is best possible by an example, but leaves the bound open.