The classification conjecture for 3-chromatic distances in the three-dimensional integer lattice

Let $m$ be a positive integer, and let $\chi(\mathbb{Z}^{3},\sqrt{m})$ denote the minimum number of colours needed to colour $\mathbb{Z}^{3}$ so that points at distance $\sqrt{m}$ receive different colours. Classification conjecture. There are no other examples of chromatic number $3$: $\chi(\mathbb{Z}^{3},\sqrt{m})=3$ only for those $m$ that can be represented as

$$2^{2k}l,$$

where $l\equiv 10\pmod{12}$. The conjecture also proposes that chromatic number $3$ never occurs for integer lattices in dimensions higher than three. This extends the established examples and leaves open the complete classification of 3-chromatic distances in $\mathbb{Z}^{3}$ and the higher-dimensional question.