Suppose $(X_1, X_2), (Y_1, Y_2)$ are two random vectors in $\mathbb{R}^2$ with the same (joint) distributions and suppose $\mathbb{E}[X_1|X_2]=0$ and $\mathbb{E}[X_1^2|X_2]=1$ and $\mathbb{E} X_2^4<\infty.$ Let $f:\mathbb{R}\to\mathbb{R}_{\geq 0}$ be a non-negative function such that $\mathbb{E}\left[\left(X_1\cdot f(X_2)\right)^2\right]<\infty.$ Can we now show that $$\mathbb{E}[|f(X_2)X_1X_2Y_1Y_2|]<\infty?$$
And if this is not true, what if $(X_1, X_2), (Y_1, Y_2)$ are two elements of a strongly mixing, strictly stationary sequence of random variables?
I have tried the following, using repeated applications of Cauchy-Schwarz: $$\begin{align*} \mathbb{E}[|f(X_2)X_1X_2Y_1Y_2|]&\leq \sqrt{\mathbb{E}[(X_1f(X_2))^2]\cdot \mathbb{E}[Y_1^2Y_2^2X_2^2]}\\ &\leq \sqrt{\mathbb{E}[(X_1f(X_2))^2]\cdot \sqrt{\mathbb{E}[Y_1^2Y_2^4]\cdot\mathbb{E}[Y_1^2X_2^4]}}. \end{align*}$$
By assumption, $\mathbb{E}[(X_1f(X_2))^2]$ is finite and we have that $\mathbb{E}[Y_1^2Y_2^4] = \mathbb{E}[\mathbb{E}\left[Y_1^2|Y_2]\cdot Y_2^4]\right]=\mathbb{E}[Y_2^4]<\infty.$ However, it seems to me that $\mathbb{E}[Y_1^2X_2^4]$ need not necessarily be finite, since we need not have $\mathbb{E}[Y_1^2|X_2]=1$. Could a different approach perhaps work, or could we somehow use the fact that $(X_1, X_2), (Y_1, Y_2)$ come from a strictly stationary and strongly mixing sequence?