  We show that for $\varepsilon > 0$, every $C^{1 + \varepsilon}$ skew product
on $\mathbb{T}^2$ over a rotation of $\mathbb{T}^1$ satisfies Sarnak's
conjecture. This is an improvement of earlier results of
Kulaga-Przymus-Lema\'nczyk, Huang-Wang-Ye, and
Kanigowski-Lema\'nczyk-Radziwill.
