Can we always write a random variable as conditional expectation plus independent error? Consider the random variables $Y,X$. Can we always write
$$
 Y=E(Y|X)+\epsilon\tag 1
$$
with $\epsilon$ independent of $X$?
Note: from this answer here, we know that we can always write (1) with $E(\epsilon|X)=0$. However, here I am asking if we strengthen the relationship between $\epsilon $ and $X$ to stochastic independence.

Suppose
$$
Y=X^2+u
$$
where $u|X\sim(0,X^2)$ has conditional heteroskedasticity. Then,
$$
\epsilon=Y-E(Y|X)=X^2+u-E(Y|X)=u,
$$
which has conditional mean zero but is not independent of $X$, as its second moment depends on $X$.
