If we have quenched decay of correlation, can we transfer it to annealed decay of correlation? To be precise, let us consider following setting:

Given transformations $T_{\omega}: (S^1, dm) \to (S^1, dm)$ indexed by $\omega \in (\Omega, \sigma, \mathbb{P})$ where $\mathbb{P} $ is ergodic (or Bernoulli) probability on $\Omega$ and $dm $ is Leb measure on circle $S^1$. define random composition $T_\omega^n:=T_{\sigma^{n-1}\omega} \circ \dots \circ T_\omega$. Assume we have quasi-invariant absolutely continuous probability $\mu_\omega:= h_\omega\, dm$ such that $(T_\omega)_{*} \mu_\omega=\mu_{\sigma \omega}, h_\omega \in \operatorname{Lip}(S^1)$.

So we have invariant transformation $\tau $ defines on skew product space $(\Omega \times S^1, \mu)$:

$$ \tau (\omega, x): =(\sigma \omega, T_{\omega}x) \text{ where } \mu = d\mu_{\omega} \, d\mathbb{P} \text{ is invariant probability on } \Omega \times S^1. $$

we study two decay of correlations:

$\forall \phi, \psi \in L^\infty(\Omega \times S^1)$ and $\psi_\omega \in \operatorname{Lip}(S^1)$ for all $\omega \in \Omega$

Quenched decay of correlation:

$$ \left|\int \varphi_{\sigma^n \omega} \circ T^n_\omega \cdot \psi_\omega \, d\mu_\omega -\int \varphi_{\sigma^n \omega} \, d\mu_{\sigma^n \omega} \int \psi_\omega \, d\mu_\omega\right| \le C_{\omega} \cdot \|\varphi\|_\infty \cdot C_\psi \cdot e^{-n},$$ where $\mathbb{P}(C_\omega >m) \le \frac{1}{m^2}$

Anneal decay of correlation:

$$\left| \int \varphi \circ \tau^n \cdot \psi \, d\mu-\int \varphi \, d\mu \cdot \int \psi \, d\mu \right| \le \|\varphi\|_\infty \cdot C_\psi \cdot e^{-n} $$

I tried to prove Quenched decay of correlation $\implies $ Annealed decay of correlation, but got stuck because $\psi-\int \psi \, d\mu$ is no longer fiber-wise mean zero. I guess we should assume certain mixing on $\sigma$.

is it really true or has counter-example? Thanks in advanced.