Stretched-exponential mixing for $\mathscr{C}^{1+\alpha}$ skew products with discontinuities

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