Does independence of conditional expectations imply conditional independence?

Question

Suppose I have $E(X|\mathcal{F}) \perp E(Y|\mathcal{F})$, is this sufficient to show that $X$ is conditionally independent of $Y$ given $\mathcal{F}$?

This answer: https://math.stackexchange.com/a/2660070/82654 shows that the converse is false, so I am wondering if one is strictly stronger

Answer 1

Turns out this is trivially false. Take $Z$ to be some random variable independent of $X$ and let $Y = X$, then we have $E(X|Z) = EX = E(Y|Z)$ and so $E(X|Z)$ is independent of $E(Y|Z)$ (since they are both constants), but obviously $X$ is not independent of itself for any non-degenerate $X$.