Absolute continuity and exponent rigidity for large measures

Let $M$ be an infranilmanifold, let $\alpha_0$ be a totally non-symplectic $\mathbb{Z}^k$ action by automorphisms of $M$, and let $\alpha$ be an action with homotopy data $\alpha_0$. Call an $\alpha$-invariant Borel probability measure large when its pushforward under the semiconjugacy to the algebraic action is Haar measure. Large-measure rigidity conjecture. Every large invariant measure for $\alpha$ is absolutely continuous and has the same Lyapunov characteristic exponents as $\alpha_0$. The claim is proposed as the desired extension to the resonant setting; the source says that the relevant proofs will appear in a subsequent paper and gives no resolution here.