How does the 4-color theorem generalize for spaces of higher-dimensions and non-zero curvature described by Coexter-Dynkin diagrams?

Are there known upper and lower bounds for the number of colors required for drawing a multi-dimensional tesselation described by a Coxter-Dynkin diagram?

By skimming through “Higher-Dimensional Analogues of the Map Coloring Problem” by Bhaskar Bagchi and Basudeb Datta I assume that the upper-bound number of colors given the number of dimensions is 3^n+3^(n-1)-2

Nonetheless, I’m not sure how the curvature of the space influences this.
Do you have any other suggestions or approaches?

How does the 4-color theorem generalize for spaces of higher-dimensions and non-negative curvature described by Coexter-Dynkin diagrams?

I have also posted a similar question on Reddit.

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