The multivariate mean equicontinuity characterization of finite topomorphic extensions

Let $(X,\varphi)$ be a minimal topological dynamical system, and let $(Y,\psi)$ be its maximal equicontinuous factor (MEF). For $m\in\mathbb{N}$, say that $(X,\varphi)$ is mean $m$-equicontinuous if, for every $\varepsilon>0$, there exists $\delta>0$ such that sufficiently close $m$-tuples have Besicovitch $m$-distance less than $\varepsilon$. Say that $(X,\varphi)$ is an $m$:1 topomorphic extension of its MEF if $m$ is the least integer for which there is a measurable choice of at most $m$ points in almost every fibre of the factor map to $(Y,\psi)$, with full measure for every $\varphi$-invariant measure. The multivariate mean equicontinuity conjecture. $(X,\varphi)$ is mean $(m+1)$-equicontinuous but not mean $m$-equicontinuous if and only if it is an $m$:1 topomorphic extension of its MEF. The known implication is that every minimal $m$:1 topomorphic extension is mean $(m+1)$-equicontinuous. For $m=1$, the corresponding equivalence is known, whereas the converse in the multivariate case is stated as a conjecture and is not pursued in the paper.