Stretched-exponential mixing for skew products with discontinuities

Consider the skew product $F:\mathbb{T}^{2}\rightarrow \mathbb{T}^{2}$, $F(x,y)=(f(x),y+\unicode[STIX]{x1D70F}(x))$, where $f:\mathbb{T}^{1}\rightarrow \mathbb{T}^{1}$ is a piecewise $\mathscr{C}^{1+\unicode[STIX]{x1D6FC}}$ expanding map on a countable partition and $\unicode[STIX]{x1D70F}:\mathbb{T}^{1}\rightarrow \mathbb{R}$ is piecewise $\mathscr{C}^{1}$. It is shown that if $\unicode[STIX]{x1D70F}$ is not Lipschitz-cohomologous to a piecewise constant function on the joint partition of $f$ and $\unicode[STIX]{x1D70F}$, then $F$ is mixing at a stretched-exponential rate.