The tropical equilateral-dimension conjecture

Let $d4a3^n$ be equipped with the tropical metric, and let $\mathrm{e}_{\mathrm{tr}}(\mathbb{R}^{n})$ denote the maximal cardinality of a tropical equilateral subset of $d4a3^n$. The tropical equilateral-dimension conjecture.

$$\mathrm{e}_{\mathrm{tr}}(\mathbb{R}^{n})=\binom{n+1}{\lfloor (n+1)/2\rfloor}.$$

The lower bound is attained by an explicit equilateral set in $\mathbb{Z}^n$, and equality is proved in dimensions $2$ and $3$ and for points on a tropical sphere at pairwise tropical distance $2R$. The general conjecture remains open.