Sphere coloring conjecture for discrete spheres

Let a $d$-sphere be a finite simple graph defined recursively by requiring every unit sphere to be a $(d-1)$-sphere and requiring the deletion of some vertex to be contractible; the empty graph is the $(-1)$-sphere. The chromatic number of a graph is the least number of colors in a proper vertex coloring.

Sphere coloring conjecture. Every $d$-sphere has chromatic number $d+1$ or $d+2$.

This generalizes the characterization of the four-color theorem in terms of $2$-spheres and proposes the corresponding dimensional bound. The source presents it as an open conjecture, with no proof known in the stated generality.