Sharpness conjecture for the three-uniform Erdős–Frankl–Pach bound

Let $M_3(n)$ denote the maximum size of a family $\mathcal{F}\subseteq\binom{[n]}{4}$ with $\operatorname{VC}(\mathcal{F})\le 3$. The paper establishes the lower bound

$$M_3(n)\ge \binom{n-1}{3}+n-3$$

for every $n\ge 6$. Three-uniform sharpness conjecture. For all sufficiently large $n$,

$$M_3(n)=\binom{n-1}{3}+n-3.$$

This is presented as a suspected sharp bound; the restriction to sufficiently large $n$ is necessary because $M_3(8)\ge45$ while the displayed expression equals $40$.