Bound on the number of unit distances in higher-dimensional integral point sets

Let $M\in\mathfrak{M}(m,n)$ and let $M_1,M_2\in M$ satisfy $|M_1M_2|=1$. A blowup of a facher set is the construction referred to by the source, and a unit distance is said to occur once for each pair of points at distance $1$. Unit-distance-count conjecture. Either $n=m+1$, or $M$ is a blowup of a facher set and distance $1$ occurs in $M$ no more than

$$1+\frac{(m-1)^2-(m-1)}{2}$$

times. The source states that the planar case is already known and proposes this higher-dimensional estimate based on the number of edges in an $(m-2)$-dimensional simplex; the general assertion remains open.