Atomic approximation conjecture for biparameter martingale filtrations

Let $X$) be a separable Banach space. For integers $N,M\geq 0$, let $(f_{n,m}:0\leq n\leq N,0\leq m\leq M)$ be an $X$-valued martingale on $(\Omega,\mathcal{F},\mathbb{P})$ with respect to a filtration $(\mathcal{F}_{n,m}:0\leq n\leq N,0\leq m\leq M)$ satisfying the

-condition. An **atomic filtration** is a filtration whose members are atomic \sigma-algebras. The conjecture asserts that, for every $\varepsilon>0$, there \exists an atomic filtration $(\mathcal{G}_{n,m}:0\leq n\leq N,0\leq m\leq M)$ satisfying the

-condition such that

$$\left\|f_{n,m}-\mathbb{E}\left(f_{N,M}\mid\mathcal{G}_{n,m}\right)\right\|<\varepsilon$$

for all $0\leq n\leq N$ and $0\leq m\leq M$.