The pro-nilsystem conjecture for distal minimal topological dynamical systems

Let $(X,T)$ be a distal minimal topological dynamical system, and let ${\bf Ind}^{[d]}_{ap}(X,T)$ denote its arithmetic-progression independence relation of order $d$. Let $\Delta(X)$ be the diagonal of $X\times X$. Pro-nilsystem conjecture. If

$${\bf Ind}^{[d]}_{ap}(X,T)=\Delta(X),$$

then $(X,T)$ is a $d$-step pro-nilsystem. This would strengthen the preceding result that the order of $(X,T)$ is less than $d$ under the same hypothesis; the source does not provide a resolution.