If $Cov(X,Y)=0$ and $Cov(X,Y|Z)=0$, does it mean $Cov(Y,Z)=0$?

Question

Let $X,Y,Z$ be $L^2$ random variables, not constants.

If $Cov(X,Y)=0$ and $Cov(X,Y|Z)=0$, does it mean $Cov(Y,Z)=0$ ?

My opinion is no, but I cannot find a counterexample.

$Cov(X,Y)=0$ and $Cov(X,Y|Z)=0$ imply that $Cov(E[X|Z],E[Y|Z])=0$, which is far away from $Cov(Y,Z)=0$.

Answer 1

If it works for a constant, it's almost always still going to work if you require them to be non-constant. For another example, here you can take $Y=Z$ and $X$ independent of $Y$.