Is it true that $f(n)\leq \frac{n}{2}+O(1)$?

OPEN

Let $f(n)$ be maximal such that there exists a set $A$ of $n$ points in $\mathbb{R}^4$ in which every $x\in A$ has at least $f(n)$ points in $A$ equidistant from $x$.

Is it true that $f(n)\leq \frac{n}{2}+O(1)$?

Erdős, Makai, and Pach proved that
\[\frac{n}{2}+2 \leq f(n) \leq (1+o(1))\frac{n}{2}.\]

See also [753].