Is the relation E[E[X|Z]|Y ] = E[X|Z] true or False? 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 E[A] if A and Y are independent.
I would like to know if my thought process is accurate or not

No.
For a counterexample let $Z=X$ and let it be that $X$ is not measurable wrt $Y$.
Then $\mathbb E[X\mid Z]=X$ so that $\mathbb E[\mathbb E[X\mid Z]\mid Y]=\mathbb E[X\mid Y]$ but also $X\neq\mathbb E[X\mid Y]$.
It can at most be stated that both rv's have the same expectation as $X$:$$\mathbb E[\mathbb E[\mathbb E[X\mid Z]\mid Y]]=\mathbb E[\mathbb E[X\mid Z]]=\mathbb EX$$where both equalities are based on the general rule: $$\mathbb E[\mathbb E[U\mid V]]=\mathbb EU$$
