Equality of arithmetic-progression and regionally proximal relations for minimal distal systems

Let $(X,T)$ be a minimal distal system. The arithmetic-progression proximality conjecture.

$${\bf AP}^{[d]}={\bf RP}^{[d]} \quad\text{for every } d\in\mathbb N.$$

Here ${\bf AP}^{[d]}$ denotes the regionally proximal relation of order $d$ along arithmetic progressions, while ${\bf RP}^{[d]}$ denotes the regionally proximal relation of order $d$. The conjecture would identify these two characteristic relations for all minimal distal systems and connect the arithmetic-progression approach with the established theory of pro-nilfactors; its status is not resolved in the supplied source.