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We compute the set of facets of the polytope which is the convex hull of theCoxeter 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 facetswhich provides a counterexample to the conjecture of \cite{birkhoff}.

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