For the random variables X,Y and Z, Is the expression E[E[X|Z]|Y] = E[X|Z] true or False given we do not know if X,Y,Z are independent or not.

Question

Given, we do not know if $X,Y,Z$ are independent or not. Will the expression $E[E[X|Z]|Y] = E[X|Z]$ hold true?

What I have tried:

Now, $E[X|Z]$ is a random variable so let $E[X|Z]$ be $A$

So, problem boils down to $E[A|Y]$ which can only be equal to $A$ if A and Y are independent.

I would like to know if my thought process is accurate or not

Answer 1

Just choose $X=Z$. In this case $E[X|X]=X$ and therefore:

$E[E[X|Z]|Y]=E[X|Y]$

$E[X|Z]=X$

but in general $E[X|Y]$ is different from $X$.