The Markov-partition conjecture for the Jeandel–Rao toral rotation

Let $\mathcal{P}_0$ be the toroidal partition associated with the Jeandel–Rao Wang shift, let $\Gamma_0$ be the corresponding lattice, and let $R_0$ denote the associated $\mathbb{Z}^2$-rotation on $\mathbb{R}^2/\Gamma_0$. A partition is a Markov partition for this system when it has the Markov property with respect to the $\mathbb{Z}^2$-action and rotation. Markov-partition conjecture. The partition $\mathcal{P}_0$ is a Markov partition for

$$(\mathbb{R}^2/\Gamma_0,\mathbb{Z}^2,R_0).$$

The conjecture would complete the identification of the minimal subshift $X_0$ with $\mathcal{X}_{\mathcal{P}_0,R_0}$ and establish the stated symbolic coding as a Markov partition for the toral $\mathbb{Z}^2$-rotation. The source says that this equality and the resulting statement are deferred to later work, so the conjecture is presented as open.