Measure-zero conjecture for the nonminimal Jeandel–Rao tilings

Let $\Omega_0$ be the Jeandel–Rao Wang shift and let $X_0\subset\Omega_0$ be the minimal aperiodic subshift constructed in the paper. A shift-invariant probability measure on $\Omega_0$ is a probability measure invariant under the shift action. Measure-zero conjecture.

$$\Omega_0\setminus X_0 \text{ is of measure zero for any shift-invariant probability measure on }\Omega_0.$$

This conjecture asserts that the constructed minimal subshift gives an almost complete description of the Jeandel–Rao tilings from the measure-theoretic viewpoint. The paper notes that the complement is nonempty because of horizontal fault lines, but does not establish the claimed measure-zero property.