The spherical coloring conjecture for the 2-sphere

Let $\mathbb{S}^2$ be the unit $2$-sphere. Spherical coloring conjecture. There exists a map $q:\mathbb{S}^2 \rightarrow \{\pm 1, \pm 2, \pm 3, \pm 4\}$ such that antipodal points of $\mathbb{S}^2$ receive opposite values, and any three points equidistant on a great circle have values summing to zero. The supplied text presents this as one of two conjectures intended to imply Tutte's $5$-flow conjecture; its status is not resolved there.