Does $\Bbb E[X|Z]=\Bbb E[Y|Z]$ if $X,Y$ are identically distributed random variable?

Question

Does $\Bbb E[X|Z]=\Bbb E[Y|Z]$ if $X,Y$ are identically distributed random variable, where $Z$ is a third random variable? Thank you!

Answer 1

Not necessarily. Example: $Z=X$. In this case $\Bbb E[X|Z] = Z$ but $\Bbb E[Y|Z]=E[Y]$.