The coloring conjecture for the rhombus $\rho_\lozenge$

Let $\mathcal R^3$ be the unit distance graph of $\mathbb R^3$, and let $\rho_\lozenge=[uvwx]\subset\mathbb R^3$ be the unit rhombus defined in the source. A coloring is a map $\chi:\mathcal R^3\to\{1,2,3\}$. Coloring conjecture for $\rho_\lozenge$. There is a coloring $\chi$ with no rainbow unit triangles and with $\chi(u)=\chi(v)=1$, $\chi(w)=2$, and $\chi(x)=3$. Such a coloring would imply that $\rho_\lozenge$ cannot be domed by the stated proposition.