If random variables A and B are conditionally independent given X and variables A and C are conditionally independent given X, does it follow that variables B and C are independent given X? I don't think that's true but I can't come up with a counterexample.
Counterexample: Choose your favorite $A$ and $B$ that are conditionally independent given $X$ and set $C=B$. $B$ is not, in general, conditionally independent of itself given $X$. (This fails if $A$ is some deterministic function of $X$.)
Note the analogous non-conditional result is not even true: if $A$ and $B$ are independent, and $A$ and $C$ are independent, it does not necessarily follow that $B$ and $C$ are independent.