The tangent-measure conjecture for spectral measures of sparse decaying Jacobi matrices

Let $J$ be a sparse decaying Jacobi matrix with spectral measure

$\mu$. **Tangent-measure conjecture.** For every

$\xi\in(-2,2)$, the spectral measure

$\mu$ has a unique tangent measure at

$\xi$, and this tangent measure is Lebesgue measure.

Zlatoš proved that sparse decaying Jacobi matrices have one-dimensional spectral measures on $(-2,2)$, while the cited corollary gives more precise local behavior for the narrower class considered by Breuer. The conjecture asks whether the stronger uniqueness and Lebesgue tangent-measure statement holds for every sparse decaying Jacobi matrix.