Conditional expectation with mismatched densities

Question

Suppose, we have r.v. $X,Y,Z$. Let $X$ and $Z$ be independent. Suppose all of the variables have zero mean and finite second moment.

Next, suppose that $E_Y[(E_{X|Y}[X|Y])^2]< \infty$ and is defined as \begin{align*} E_Y[(E_{X|Y}[X|Y])^2] = \int (E_{X|Y}[X|Y=y])^2 f_Y(y) dy \end{align*} What can we say about conditional expectation $E_Z[(E_{X|Y}[X|Y=Z])^2]$ defined as \begin{align*} E_Z[(E_{X|Y}[X|Y=Z])^2]=\int (E_{X|Y}[X|Y=z])^2f_Z(z) dz \end{align*} Is it zero since $X$ and $Z$ are independent? Or is it just $(E_{X|Y}[X|Y=z])^2$ averaged over a wrong distribution?

Using Jensen's inequality and law of total expectation one can show that \begin{align*} E_Z[(E_{X|Y}[X|Y=Z])^2] \le E_Z[E_{X|Y}[X^2|Y=Z]] = E[X^2]< \infty \end{align*}

Answer 1

What can we say about conditional expectation $E_Z[(E_{X|Y}[X|Y=Z])^2]$ defined as $$E_Z[(E_{X|Y}[X|Y=Z])^2]=\int(E_{X|Y}[X|Y=z])^2f_Z(z)\mathrm dz$$

That it should be:

$$E_Z[(E_{X\mid Y=Z}[X|Y=Z])^2]=\int(E_{X|Y,Z}[X|Y=z,Z=z])^2f_Z(z)\mathrm dz$$

We're told that $X$ is independent of $Z$, but we don't know if $X$ is conditionally independent of $Z$ when given $Y$.