The zero mean dimension conjecture for the marker property

An aperiodic dynamical system is a dynamical system with no periodic points, and its mean dimension measures the dynamical dimension per iterate. The marker property means that for every $N\in\mathbb{N}$ there is an open set whose first $N$ iterates are pairwise disjoint and whose iterates cover the whole space.

Zero mean dimension conjecture. Every aperiodic dynamical system with zero mean dimension has the marker property.

Gutman proved that every finite-dimensional aperiodic dynamical system has the marker property, so the conjecture reduces to aperiodic systems of zero mean dimension that are infinite-dimensional. The general case remains open.