The Birkhoff polytope of the groups ${\mathsf{F}}_4$ and ${\mathsf{H}}_4$

We compute the set of facets of the polytope which is the convex hull of the Coxeter groups ${\mathsf{F}}_4$ or ${\mathsf{H}}_4$: For the group ${\mathsf{F}}_4$ we found $2$ orbits of facets which contradicts previous results published in \cite{birkhoff}. For the group ${\mathsf{H}}_4$ we found $1063$ orbits of facets which provides a counterexample to the conjecture of \cite{birkhoff}.