Conjecture on unit simplices in diameter graphs

Let $\mathrm{DIAM}(d,k,n)$ be the maximum number of unit $k$-simplices spanned by an $n$-point set in $\mathbb{R}^d$ of diameter $1$. Conjecture on unit simplices in diameter graphs. For $d\geq 2$ and $k\leq d$ we have

$$\mathrm{DIAM}(d,k,n)=O_d(\min\{n^k,n^{d-k+1}\}),$$

and, for odd $d$ and $k=(d+1)/2$, we have

$$\mathrm{DIAM}(d,k,n)=O_d(n^{k-1}).$$

A theorem in the paper proves the weaker general bound $O_{d,\varepsilon}(n^{d/2+\varepsilon})$, so the conjecture is confirmed only up to a small error term in the stated setting. The sharper estimates remain open.