Algebraic characterization of the smallest closed invariant equivalence relation containing the higher order regionally proximal relation

Let $(X,T)$ be a minimal system, let $d 1$ be an integer, and let $G$, $M$, $D$, $J(M)$, and $H_{\uptau^{[d]}}(G)$ be the groups and spaces defined in the paper. Write $\bold{RP}^{[d]}(X)$ for the higher order regionally proximal relation, and let $\mathcal{A}(\bold{RP}^{[d]}(X))$ denote the smallest closed invariant equivalence relation containing it.

The conjecture.

$$\mathcal{A}(\bold{RP}^{[d]}(X))=\{(x,vhgx):x\in X,\ h\in H_{\uptau^{[d]}}(G),\ g\in D,\ v\in J(M)\}.$$

This would give an algebraic description of the maximal factor of order $d$ for arbitrary group actions, and in particular would imply consequences for higher order regionally proximal relations under the Bronstein condition. The paper presents the statement as unresolved; proving it is complicated by the failure of right multiplication and inversion to be $\uptau^{[d]}$-homeomorphisms and by the structure of cube spaces when $d>2$.