Global rigidity of abelian Anosov algebraic actions

As part of a more general conjecture, it was asked if all smooth Anosov $Z^r$-actions without rank-1 factor on tori and nilmanifolds are, up to smooth conjugacy, actions by automorphisms. Recently, Fisher, Kalinin and Spatzier showed this holds under the extra assumption that the action has at least one Anosov element in every Weyl chamber of the linearization action. We will verify that this assumption always holds, hence fully establish the statement above. This is a joint work with Federico Rodriguez Hertz.