NOTES

New approach for the 5-cells:

Pick a tetrahedron of an inscribed 600-cell with vertices A, B, C, D

This gives pairs of vertices:

AB AC AD BC BD CD

Each of these gives rise to seven pairs of 5-cells which are on neighboring vertices of the 5 600-cells.

Try enumerating these and inspecting them to find one or more coherent sets of four 5-cells which lie on one tetrahedron from each of the 600-cells.

(I expect there to be more than one, like how there are two ways to partition the 120-cell vertices into 600-cells)