Suppose I have a random variable $Z$ defined on a probability space built on a $\sigma$-algebra $\mathcal{F}$ (with $\mathbb{E}[Z^2] < \infty$). Moreover, suppose that I have two filtrations $\mathcal{A}$, $\mathcal{B} \subseteq \mathcal{F}$. My question is: If we define the random variables $$ X = \mathbb{E}[Z | \mathcal{A}] \text{ and } Y = \mathbb{E}[Z | \mathcal{B}], $$ when can we guarantee that $X$ and $Y$ are positively correlated?

In all the examples where I've constructed $\mathcal{F}$ by hand, it is easy to verify that $X$ and $Y$ are positively correlated. But I'm hoping that there's a generic argument out there showing that this property always holds (provided that $\mathcal{F}$ is nice in some way).