3 problems

Let $(X,T)$ be a minimal nilsystem and let $d\in\mathbb N$. For $k\geq 2$, write $X_k$ for the maximal $k$-step nilfactor of $(X,T)$, $N_d(X,T)$ for the associated orbit-closure sy…

Pro-nilfactor conjecture. If $(X,T)$ is a distal minimal system, then its maximal $(d-1)$-step pro-nilfactor is its topological characteristic factor along arithmetic progressions…

Distality conjecture. The hypothesis that $(X,T)$ is distal in the theorem asserting that ${\bf RP}^{[d]}$ is a closed invariant equivalence relation and that the quotient by this…