Measure uniqueness and measurable rigidity for Cartan actions on tori

Let $lpha$ be the action on $\mathbb T^{k+1}$, let $lpha_0$ be its algebraic factor, and let $h$ be the semiconjugacy from $lpha$ to $lpha_0$. Let $\lambda$ denote Lebesgue measure on $\mathbb T^{k+1}$, and let $\mathcal M$ be the set of $lpha$-invariant Borel probability measures $nu$ satisfying $(h)_*nu=\lambda$.

Measure uniqueness and measurable rigidity conjecture. The set $\mathcal M$ consists of a single measure, and the semiconjugacy $h$ is a measurable isomorphism between the actions $\alpha$ and $\alpha_0$.

This would strengthen the preceding result that every measure in $\mathcal M$ is absolutely continuous and that $\mathcal M$ contains at most countably many ergodic measures. It asserts both uniqueness of the invariant measure projecting to Lebesgue measure and measurable equivalence of the original action with its algebraic factor.