Convex hull of Coxeter groups

For a given Coxeter group W(S) acting on Rn, we can view it as a set of vector in Mn(R).
Then we can consider the facets of this polytope and their enumeration
This was considered in this paper where it was proved that for An and Bn there is only one orbit of facets given by the Birkhoff tensors.
For F4 it was claimed that there is only one orbit but this is false
For H4 we found 1063 orbits of facets. See this document