Hölder-continuous Möbius disjointness for nonzero-degree skew products

Determine whether Sarnak's conjecture holds for the skew product T_{\alpha,h}(x,y)=(x+\alpha,y+h(x)) when \alpha is irrational, h is Hölder continuous, and \deg(h)\ne 0, extending the known result for Lipschitz-continuous h.

Background

The paper proves Sarnak's conjecture for certain degree-zero skew products satisfying a quantitative deviation condition, as well as for several classes of rigid pseudo-rotations. For a skew product with \deg(h)\ne 0, however, the rotation vector is not well-defined, so the paper's rigidity theorem for skew products does not apply.

Prior work established Sarnak's conjecture for every irrational rotation parameter when h is Lipschitz continuous. The unresolved issue is whether that conclusion remains valid under the weaker assumption that h is only Hölder continuous.