Veech's conjecture on the Pinsker orthocomplement of Möbius orbit measures

Let $X_{cmathcal{A}_3}$ be the three-symbol sequence space, let $S$ be the shift, let $\bm \mu$ be the Möbius sequence, and let $\mathcal{I}_S(\bm \mu)$ denote the set of $S$-invariant measures arising as limit points of the empirical measures of the orbit of $\bm \mu$. For $\eta\in\mathcal{I}_S(\bm \mu)$, let $\Pi_\eta(S)$ be the Pinsker algebra and let $\operatorname{pr}_1$ be the first-coordinate function.

Veech's conjecture. For any $\eta\in\mathcal{I}_S(\bm \mu)$,

$$\operatorname{pr}_1\in L^2(X_{\mathcal{A}_3},\Pi_\eta(S),\eta)^\perp.$$

This conjecture is attributed to W. Veech and is used to control the singular component of the Möbius flow's spectrum; its resolution status is not specified in the source.