I am embarrassed to be stuck on this seemingly simple question.

Suppose that $X,Y$ are mean-zero real-valued random variables and $\tilde X,\tilde Y$ are their "independent copies": $\tilde X,\tilde Y$ are mutually independent and independent of $(X,Y)$, and $\tilde X$ (resp., $\tilde Y$) is distributed identically to $X$ (resp., $Y$).

Here is the inequality I am trying to prove/disprove:
for some universal constant $c>0$,
$$
\mathbb{E}|\tilde X-\tilde Y|
\le
c\left(
\mathbb{E}|X-Y|
+
\sqrt{|\mathbb{E}XY|}
\right).
$$

Update. Note that the related inequality,
$$
\mathbb{E}|\tilde X-\tilde Y|^2
\le
\mathbb{E}|X-Y|^2
+
|\mathbb{E}XY|
,
$$
is trivially true.