Suppose that X,$Z_1 ... Z_n$ have a multivariate normal distribution, and X has zero mean. Furthermore, suppose that $Z_1 ... Z_n$ are independent. Show that
$$\mathbb{E}[X|Z_1 ... Z_n] = \sum_{i=1}^n \mathbb{E}[X|Z_i]$$
Is this result true without the multivariate normal condition? Prove or give a counter example.
I am able to show the first part where the multivariate normal condition is used. I am however unsure about the result without the multivariate normal condition.
Take $n=2$, $Z_1$ and $Z_2$ i.i.d. with standard normal distribution and $$ X=\left(\mathbf{1}_{Z_1>0}-\mathbf{1}_{Z_1\leqslant 0}\right)Z_2. $$ Then $X$ has a standard normal distribution, $(X,Z_1,Z_2)$ is not a Gaussian vector, $\mathbb E\left[X\mid Z_1\right]=\mathbb E\left[X\mid Z_2\right]=0$ but $\mathbb E\left[X\mid Z_1,Z_2\right]=X$.
This example is inspired from one of the most classical one providing a non-Gaussian vector whose marginals are Gaussian.