Law of total expectation and conditional expectation

Question

We know from law of total expectation that $$ \mathbb{E}[\mathbb{E}[Y|X]]=\mathbb{E}[Y] $$ Does that still work if there is a further condition, i.e. does this equation hold? $$ \mathbb{E}[\mathbb{E}[Y|X]|Z=z]=\mathbb{E}[Y|Z=z] $$

Answer 1

No, it does not. A counterexample would be $Y=Z$, $X$ independent of $Y$. Then $ \mathbb{E}[\mathbb{E}[Y|X]] = \mathbb{E}Y$, $$\mathbb{E}[\mathbb{E}[Y|X]|Z=z] = \mathbb{E} Y,$$ $$\mathbb{E}[Y|Z=z] = z.$$

Of course, we need $Y$ to be non-degenerate.