Suppose $x$ and $y$ are random variables with positive variance. Let $\rho_{x,y} \in [-1,1]$ denote their (Pearson) correlation coefficient. Let $\mathbb{E}(x\mid y)$ denote the conditional expectation of $x$ given $y$ and similarly $\mathbb{E}(y\mid x)$ is the conditional expectation of $y$ given $x$. Construct two functions: $$\lambda(x) \equiv x + \mathbb{E}(y\mid x) \text{ and } \mu(y) \equiv y + \mathbb{E}(x\mid y).$$ We can thus think of $\lambda$ and $\mu$ themselves as random variables. Assume the variances of both $\lambda$ and $\mu$ are strictly positive. Let $\rho_{\lambda,\mu}$ be the correlation coefficient for these random variables.
Conjecture: $\rho_{x,y} \le \rho_{\lambda,\mu}$.
I've spent some time trying to prove this. It is easy to show when the conditional expectations are linear, e.g. when $x$ and $y$ are jointly normal. In that case, $|\rho_{x,y}| = |\rho_{\lambda,\mu}|$. I've also used Monte Carlo to construct many examples where $x$ and $y$ are discrete. I have not found a counterexample. I feel like this must be a known result, but my google skills have failed me. Any ideas are appreciated.