Classification conjecture for fault-line tilings in the Jeandel–Rao shift

Let $\Omega_4$ be the Wang shift under consideration, let $X_4\subset\Omega_4$ be the associated subshift, and let $Y_i^+$ and $Y_i^-$ be the horizontal-shift orbit closures of the tilings $y_i^+$ and $y_i^-$ for $i\in\{0,1\}$. Write $\overline{Y_i^-\odot^2Y_i^+}^\sigma$ for the corresponding shift orbit closure, and let $y_0,y_1,y_2,y_3$ be the tilings specified in the source. Fault-line classification conjecture. Every tiling in $\Omega_4$ has at most one horizontal fault line; if it has one, that fault line consists of $0$'s or $1$'s, and

$$\Omega_4\setminus X_4=\left(\overline{Y_0^-\odot^2Y_0^+}^\sigma\cup\overline{Y_1^-\odot^2Y_1^+}^\sigma\right)\setminus\overline{\{y_0,y_1,y_2,y_3\}}^\sigma.$$

Here $y_2=\jmath\,\eta\,\omega_6\,\omega_7\,\omega_8\,\omega_9\,\omega_{10}\,\omega_{11}\,\rho(z_2)$ and $y_3=\jmath\,\eta\,\omega_6\,\omega_7\,\omega_8\,\omega_9\,\omega_{10}\,\omega_{11}\,\rho(z_3)$, with $z_2$ and $z_3$ as specified in the source. The statement classifies the complement of $X_4$ as fault-line tilings, apart from the indicated exceptional orbit closure; its resolution is not established by the supplied text.