Rigidity of Jacobians along Lyapunov foliations

Let $\alpha$ be an action satisfying the assumptions of the Main Theorem, and let its Lyapunov foliations be the foliations tangent to the Lyapunov distributions. A Jacobian along such a foliation is called rigid when its logarithm is measurably cohomologous to the corresponding Lyapunov exponent.

Jacobian-rigidity conjecture. Jacobians along Lyapunov foliations for $\alpha$ are rigid.

The conjecture would provide the cocycle rigidity needed to derive measure rigidity without the special time changes used in the paper. The authors state that they are unable to prove this in their general setting, even for smooth cocycles.