The kissing-number equality for finite unit-distance configurations

For a bounded set $K\subset\mathbb{R}$, let

$$n_d(K):=\max\{\#X:X\subset\mathbb{R}^d\text{ and }|x-y|\in K\text{ for all distinct }x,y\in X\}.$$

Let $\mathrm{kiss}_d$ be the kissing number in $\mathbb{R}^d$. Kissing-number conjecture. For every dimension $d$,

$$n_d([1,2])=1+\mathrm{kiss}_d.$$

The evident lower bound comes from a central point together with a kissing configuration. The conjecture asserts that no larger finite configuration can have all pairwise distances in $[1,2]$; the source provides no proof or status beyond posing it.