Isometry overlap bound for finite planar point sets

Let ${\mathcal P}\subset\mathbb R^2$ be a finite point set. An isometry is a distance-preserving map $\varphi:\mathbb R^2\to\mathbb R^2$. The isometry overlap conjecture. There is an absolute constant $C$ such that

$$\#\{\varphi:\mathbb R^2\to\mathbb R^2\text{ isometry}\;;\;|\varphi({\mathcal P})\cap{\mathcal P}|\ge 2\}\le C|{\mathcal P}|^3.$$

The statement asks for a uniform incidence bound for congruent copies of a finite planar point set. The source explicitly presents this case as unknown; no resolution is supplied here.