On the number of regular simplices in odd dimensions

Let $S^k_d(n)$ denote the maximum number of regular $(k-1)$-simplices spanned by $n$ points in $\mathbb{R}^d$. For all fixed $r\geq k\geq3$, the exact value of $S^k_{2r}(n)$ was recently determined by Dumitrescu and the authors for all sufficiently large $n$ when $k=3$, and conditionally on an optim