This can be rephrased as follows.
Let $X$ and $Y$ be independent random variables with the same, continuous, distribution. Is it true that $E(X)\le E(|X-Y|)$.
Is this likely?
Is it true for a discrete distribution?
If one has a discrete counterexample, can one convert it to a continuous counterexample?
Added I now realise that there was some absolute value signs in the original integral (these come out badly on my screen) and I should have written $E(|X|)\le E(|X-Y|)$. Plus ca change...