Does $E[XY|Z]=E[XY]$?

Question

Assume $E[X|Z]=E[X]$. Assume $Y$ and $Z$ are independent. Does $E[XY|Z]=E[XY]$? Can you prove it?

My intuition says $E[XY|Z]=E[XY]$ but expanding the expectations into integrals I couldn't prove it.

Answer 1

No, it is not true. We will find a counterexample using a suggestion by @Did to show that $E[XY|Z] \neq E[XY]$.

Let $X=UZ$ and $Y=U$ with $U \sim Unif(-1,1)$ independent of Z, then $Y$ and $Z$ are independent and $E[X|Z]=E[UZ|Z]=ZE[U]=0$. But, $E[XY|Z]=E[ZU^2|Z]=ZE[U^2]=Z/3$ and $E[XY]=E[ZU^2]=E[Z]/3$.