Breitenbchers conjecture on mean equicontinuity and finite-to-one extensions

Let $m\in\mathbb N$, and let a minimal $\mathbb Z$-system mean a minimal dynamical system with a continuous $\mathbb Z$-action. Its maximal equicontinuous factor is the maximal factor system with an equicontinuous action, and an extension is $m$-to-one topomorphic when every fiber of the factor map has at most $m$ points and the extension satisfies the relevant topomorphic condition. Breitenb"ucher's conjecture. A minimal $\mathbb Z$-system is mean $(m+1)$-equicontinuous but not mean $m$-equicontinuous if and only if it is an $m$-to-one topomorphic extension of its maximal equicontinuous factor.

The result would characterize the precise finite multiplicity of a minimal system over its maximal equicontinuous factor through the hierarchy of mean $r$-equicontinuity. The cited work proves the forward implication from an at most $m$-to-one topomorphic extension to mean $(m+1)$-equicontinuity, while the converse is formulated here and its general status is not established in the supplied text.