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I know that independent and conditional independent don't imply each other. But what if given more condition that $Z$ is independent from $X$ and $Z$ is independent from $Y$?

So the problem is:

A: Random Variables $X$ and $Y$ are independent.

B: $X$ and $Y$ are independent given condition $Z$. $Z$ is independent from $X$ and also $Z$ is independent from $Y$.

Can B $\implies$ A be true? (Given B, can we conclude that A is true?)

Thanks for helping me prove or disprove it. I tried it by myself but only found that A is true if adding "$Z$ is also independent from $X,Y$" condition to B.

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