Pattern Sturmian spectral conjecture

Let $\bm{x}$ be a pattern Sturmian sequence, meaning that its maximal pattern complexity satisfies $p^*(n)=2n$ for all $n\geq 1$. Let $H$ be the Schrödinger operator on $l^2(\mathbb{Z})$ defined by

$$[H\psi](m)=\psi(m+1)+\psi(m-1)+\lambda x_m\psi(m).$$

Pattern Sturmian spectral conjecture. Theorem 1 holds if $\bm{x}$ is a pattern Sturmian sequence: the spectrum $\sigma(H)$ is a zero Lebesgue measure Cantor set and all spectral measures are singular continuous.

This conjecture extends the established spectral result for Sturmian sequences to the broader class of pattern Sturmian sequences. Its resolution is not specified in the source.