# Convergence in $L^2$ of iterated expectations

Take a probability space $(\Omega,\mathcal{F},\mathbf{P})$ and random variable $X \in L^2(\Omega,\mathcal{F},\mathbf{P})$.

Define the iterated expectations of X as follows: $X_0 = X$, and, inductively, $X_k = \mathbf{E}[X_{k-1} \, | \, \mathcal{G}_k]$, where $\mathcal{G}_k \subseteq \mathcal{F}$ is some $\sigma$-algebra. Assume that all the $\mathcal{G}_k$ come from some finite set of distinct $\sigma$-algebras. In words, we are repeatedly taking conditional expectations with respect to various information.

Can we conclude that the sequence $(X_k)$ converges to some limit in $L^2$?

The paper [1] says the question seems to be open (as of its publication) and reviews some of the literature, going back to Rota's Alternierende Verfahren. (The contribution of the paper is to show that if we drop the restriction of finitely many distinct $\sigma$-algebras, the statement is not true.) Has there been any progress since?

[1] M. Akcoglu and J. King. An example of pointwise non-convergence of iterated conditional expectation operators. Israel J. Math. 94 (1996). This paper was pointed out to me by Anthony Quas at a related question -- When do iterated conditional expectations converge?