if $X-Y-Z$ is a Markov chain, why $E[XZ|Y]=E[X|Y]\cdot E[Z|Y]$

Question

If I have a Markov chain $X-Y-Z$, how can I prove that $E[XZ|Y]=E[X|Y]\cdot E[Z|Y]$?

I tried to do it with Markov properties but I couldn't find it.

If $Y$ is given, does it mean that $X$ and $Z$ are independent?