Conditional random variables proof

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.