Symbolic Furstenberg conjecture for Jewett-Krieger models

Let $S,T\colon[0,1)\to[0,1)$ be the maps $Sx=px\pmod 1$ and $Tx=qx\pmod 1$, where $p,q\geq 1$ are multiplicatively independent integers. Let $\mu$ be a Borel probability measure on $[0,1)$ with Borel $\sigma$-algebra $\mathcal B$, invariant under both $S$ and $T$ and ergodic for their joint action. Let $\widehat X\subset\{0,1\}^{\mathbb Z^2}$ be a Jewett-Krieger model for the natural extension of $([0,1),\mathcal B,\mu,S,T)$. Symbolic Furstenberg conjecture. Either $\widehat X$ is finite or $\mu$ is Lebesgue measure. This reformulates Furstenberg's conjecture in terms of Jewett-Krieger models and remains open according to the supplied source context.