Prove or disprove that if $X$ is independent of $Y$, then $E[X|F]$ is independent of $E[Y|F]$

Question

If $X$ is independent of $Y$, then $E[X|F]$ is independent of $E[Y|F]$.

If the above statement is not correct, how to construct a counter example to disprove it?

Answer 1

Definitely incorrect. Let F = X+Y. Suppose X and Y are IID normal. Then E[X|F] and E[Y|F] are both linear in F, and hence perfectly correlated.