The remaining case is actually H4. For this case the conjecture is false, since I found 1063 orbits of facets. In the same paper, it is claimed that for F4 the convex hull is given by the Birkhoff tensors. But this is false since I found another orbit defined by a matrix of rank 3. See details [there][1]


  [1]: http://www.liga.ens.fr/~dutour/ConvCoxeter/index.html