I have no particular reason to think that my algorithm will always get the global minimum - it just starts with a random point set and runs a single naive convergence from there. However, I've never yet seen it give different answers on successive attempts. (It begins with the points uniformly and randomly distributed over the sphere, so if there were to be multiple significantly different solutions with catchment areas of roughly equal sizes, I'd expect to have seen one by now.)

"an octahedron with two opposing faces being 'capped' by an extra vertex"

In other words, a cube! :-)

At least, the only real difference between what you describe and a cube is that each face is potentially folded into two triangles, one part of a 'cap' and one an original face of the octahedron. I suspect that for symmetry's sake you'd want to fold the faces by little or nothing rather than by lots, otherwise you end up bringing points closer to each other than they need to be; but if you start with a cube, it appears much more likely to buckle into the shape I show, because that's quite a long way from the cube and presumably has a significant improvement in energy level.