The pro-nilfactor conjecture for topological characteristic factors along arithmetic progressions

Let $(X,T)$ be a distal minimal topological system and let $d\in\mathbb N$. The maximal $(d-1)$-step pro-nilfactor of $(X,T)$ is its topological characteristic factor along arithmetic progressions of order $d$.

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 of order $d$.

This is posed as an unsolved problem concerning the identification of topological characteristic factors for arithmetic progressions in distal minimal systems.