Relative independence of joinings with ergodic nilsystems

Let $(X,\mu,T)$ be an ergodic system satisfying the spectral hypothesis of Corollary~, and let $(Y,\nu,S)$ be an ergodic nilsystem. A joining of these systems is a joining of the measure-preserving systems $(X,\mu,T)$ and $(Y,\nu,S)$, and their Kronecker factors are their maximal rotational factors. Joining conjecture. Every joining of these two systems is relatively independent with respect to the corresponding joining of their Kronecker factors. This would extend the paper's factor results to joinings and assert that, under the spectral hypothesis, all dependence between the systems is already detected by their Kronecker factors.