Does the difference between a sequence of random variables and their conditional expectations converge in $L^1$? Let $(X_n)_n$ be a sequence of random variables on the probability space $(\Omega, \mathcal{F}, P)$, and let $(\mathcal{F}_n)_n$ be a filtration that increases to $\mathcal{F}$. We can assume $(X_n)_n$ is uniformly integrable, but I'm also interested in the general case if anyone wants to comment on that.

Is it true that $\int |X_n - E(X_n \mid \mathcal{F}_n)|dP \to 0$ as $n \to \infty$?

I haven't made any real progress on this and am just looking for some hints so I can try to prove or disprove it myself.
I know that if $X_n$ is held fixed and $\mathcal{F}_n$ is allowed to increase, then the result holds. This is just a textbook martingale convergence result. But I don't know how to generalize this to a whole sequence of random variables and puttering around with Fatou's lemma and the like hasn't gotten me anywhere.
Again, I'm just looking for some hints or suggestions so I can try to get it myself.

It's not true, and a counterexample would be one similar to my comment.  Take  $\{\xi_i\}$ i.i.d. with $P(\xi_i = 1) = P(\xi_i = - 1) = 1/2$.  Set $X_k = \prod\limits_{i = 1}^k \xi_k$ and $\mathcal{F}_n = \sigma(\{\xi_i\}_{i = 1}^{n-1})$.  Then $$|X_n - E(X_n | \mathcal{F_{n}})| = |X_n - 0| = 1.$$
This means that $E|X_n - E(X_n | \mathcal{F}_n)| = 1$.
