The tropical chromatic number conjecture for Euclidean space

Let $d4a3^n$ be equipped with the tropical metric, and let $d4a4^n$ denote its integer lattice with the induced tropical metric. The tropical chromatic number conjecture. The tropical chromatic numbers of both spaces are

$$\chi_{\mathrm{tr}}(\mathbb{Z}^{n})=\chi_{\mathrm{tr}}(\mathbb{R}^{n})=2^n.$$

The conjecture is verified in dimensions $n\leq 3$; the general case remains open.