Substep 6 is not obvious to me. It is not clear that the point where the inner sphere and the space-filling spheres intersect must be along the line from the center of the cube to the center of each sub-cube.
To put it another way, it's not obvious to me that the point of contact between the inner pink circle and the outer black circle is along the green line.
To put it another way, it's not obvious to me that the point of contact between the inner pink circle and the outer black circle is along the green line.