Does independence of conditional variable and random variable imply independence of reverse conditional variable

Question

If $X, Y, Z$ are random variables, does $X \mid Y$ and $Z$ being independent necessarily imply that $Y \mid X$ and $Z$ being independent?

Answer 1

Assuming homework:

First, you can even try make this work without the hard part, the conditionals: what if $X$ and $Y$ were independent? Then $X \mid Y = X$, and $Y \mid X = Y$. Now a counterexample should be clearer.