Projected into 3-D space, a convex 4-polytope looks like a collection of convex polyhedra. If any two convex cells sharing a face have different colors, how many colors are required? In the paper Higher dimensional analogues of the map colouring problem, it's suggested that 5 ≤ χ(3) ≤ 13 when the balls are all roughly spherical. There is also the paper Seven, eight, and nine mutually touching infinitely long straight round cylinders which shows convex cylinders with higher touching numbers. It seems these could be converted to finite convex cells, and the remaining space could be divided into convex cells that increase the required number of colors.
Is there a uniform convex 4-polytope that requires 6 or more colors? Can a polyhedron be subdivided into nine or more convex cells that all touch each other?