Prove or disprove: If $X | U$ is independent of $Y | V$, then $E[XY|U,V] = E[X|U] \cdot E[Y|V]$.

Question

I may have missed something basic here. Suppose $U$ and $V$ are continuous random variables such that $E[X|U]$ and $E[Y|V]$ makes sense for some random variables $X$ and $Y$. If $X|U$ is independent of $Y|V$, does it follow that

$$ E[XY|U,V] = E[X|U]\cdot E[Y|V]\ ? $$

Any simpler explanation is greatly appreciated.