The higher-dimensional lower-bound conjecture for ordinary hyperplanes

For $d\geqslant4$, let $e_d(n)$ denote the minimum number of ordinary hyperplanes determined by a set of $n$ points in the relevant $d$-dimensional geometric setting. Higher-dimensional lower-bound conjecture. There is a constant $c_d$ such that, for $n$ sufficiently large,

$$e_d(n)\geqslant \frac{1}{(d-1)!}n^{d-1}-c_dn^{d-2}.$$

This extends the proposed lower-bound pattern from ordinary lines and planes to dimensions at least four. The statement is presented as a conjecture, and no resolution is supplied in the source.