Incidence bound for rich affine subspaces in finite point sets

Let $C>0$ and let ${\mathcal P}\subset\mathbb R^D$ be a finite point set that is proper $D$-dimensional up to $C$. For $r<D$, let $S^{(r)}\subset\mathbb R^D$ range over affine subspaces of dimension $r$. The rich affine-subspace incidence conjecture. There is a constant $C^*$, depending on $C$, such that

$$\#\left\{S^{(r)}\subset\mathbb R^D\;;\;S^{(r)}\cap{\mathcal P}\text{ is }r\text{-dimensional up to }C\text{ and }|S^{(r)}\cap{\mathcal P}|\ge k\right\}\le C^*\frac{|{\mathcal P}|^{r+1}}{k^{D+1}}.$$

This is proposed as an incidence estimate that would simplify the study of affine mappings. The source presents it as an anticipated bound and gives no proof or resolution.