M\"obius disjointness for non-uniquely ergodic skew products

For $\tau>2$, let $T$ be a $C^\tau$ skew product map of the form $(x+\alpha,y+h(x))$ on $\mathbb T^2$ over a rotation of the circle. We show that if $T$ preserves a measurable section, then it is disjoint to the M\"{o}bius sequence. This in particular implies that any non-uniquely ergodic $C^\tau$ skew product map on $\mathbb T^2$ has a finite index factor that is disjoint to the M\"{o}bius sequence.