The Bernoulli property for $PSL_2\mathbb{R}$ skew products over exponentially mixing base

We study the Bernoulli property for skew products of the form $f(x,y)=(g(x),A(x)y)$ on $M\times PSL_2(\mathbb{R})/Γ$, where $g$ is a smooth, measure-preserving, exponentially mixing diffeomorphism, $Γ<PSL_2(\mathbb{R})$ is a cocompact lattice, and $A\colon M\to \operatorname{PSL}_2(\mathbb{R})$ is a