a paper I read states, that a Quasimartingale (an process $(X_t)_{t\in [0,T] }$ with $\mathbb E[|X_t|]<\infty$ for all $t\in [0,T]$, which suffices $\sup_\Delta \sum^{n-1}_{j=0} \left\|\mathbb E\left[X_{t_{j+1}} - X_{t_j} \middle| \mathcal F_{t_j}^X\right] \right\|_1 < \infty $ for its natural filtration $(\mathcal F^X_t)$ und $\Delta$ the set of all partitions $\pi : 0=t_0 < t_1 <\ldots < t_n = T$ of $[0,T]$) is automatically an $(\mathcal F^X_t)$-quasi-Dirichlet process, which means, that $ \sum^{n-1}_{j=0} \mathbb E \left[ \left|\mathbb E\left[ X_{t_{j+1}} - X_{t_j} \middle| \mathcal F_{t_j}^X\right] \right|^2 \right] \xrightarrow{|\pi|\to 0} 0 $ holds for $\pi : 0 = t_0 < t_1 < \ldots < t_n = T$ partitions of $[0,T]$.
While this seems reasonable, I have a hard time proving this and would be glad about any help.