Uniqueness of large invariant measures for infranilmanifold actions

Let $N$ be a simply connected nilpotent Lie group, let $A$ be a group of affine transformations acting freely with a finite-index translation subgroup that is a lattice in $N$, and let $M=N/A$ be the resulting infranilmanifold. Let $lpha_0$ be a strongly simple Anosov action of $athbb{Z}^k$ by automorphisms of $M$, and let $lpha$ be a smooth action with homotopy data $lpha_0$. Let $h:M\to M$ be the semiconjugacy satisfying $h\circ\alpha=\alpha_0\circ h$, and call an $lpha$-invariant probability measure large if $h_*$ sends it to Haar measure. Uniqueness conjecture. Under the assumptions of the theorem that every ergodic large invariant measure is absolutely continuous and has the same Lyapunov characteristic exponents as $lpha_0$, the large invariant measure is unique. The preceding theorem establishes absolute continuity and equality of exponents for an ergodic large invariant measure, while uniqueness is asserted here without a resolution in the supplied text.