Let $\{X_n\}$ and $\{Y_n\}$ be sequences of random variables s.t. $Y_n\ge 0$ a.s. and $$ \mathsf{E}[\lvert X_n-Y_n\rvert\mid \mathcal{G}_n]\to 0 \quad\text{a.s.}, $$ where $\{\mathcal{G}_n\}$ is a sequences of sub-$\sigma$-fields (not necessarily nested). Can we show that $$ \mathsf{E}[\lvert X_n\vee 0-Y_n\rvert\mid \mathcal{G}_n]\to 0 \quad\text{a.s.}? $$
In the unconditional case ($\mathcal{G}_n=\{\emptyset,\Omega\}$) it follows from the DCT (assuming that the dominating r.v. exists), i.e. $$ \mathsf{E}\lvert X_n\vee 0 - Y_n\rvert\le o(1)+\mathsf{E}\lvert X_n\vee 0-X_n\rvert\to 0. $$
Since $|X_n \lor 0 - Y_n| \le |X_n - Y_n|$ almost surely, you have $E[|X_n \lor 0 - Y_n | \mid \mathcal{G}_n] \le E[|X_n - Y_n | \mid \mathcal{G}_n]$ for each $n$.