Consider on a probability space $(\Omega,\mathcal{F},P)$ a sequence $(X_n)_n$ of random variable converging in probability to $X,(Y_n)$ is a sequence of random variable in $L^1$ converging in probability to $Y \in L^1,$ and such that for all $n \in \mathbb{N},|X_n| \leq Y_n.$ $\mathcal{Q}$ is a sub $\sigma$-algebra of $\mathcal{F}.$
Prove that if $(E[Y_n|\mathcal{Q}])_n$ converges in probability to $E[Y|\mathcal{Q}]$ then $(E[X_n|\mathcal{Q}])_n$ converges in probability to $E[X|\mathcal{Q}].$
The nonconditional case $(\mathcal{Q}=\{\emptyset,\Omega\})$ can be proved using Fatou lemma, which also holds for convergence in probability, but this way may fail in the conditional case.
Update : there are several ways to solve the problem:
by considering subsequences the problem can be reduced to a.s convergence, then we apply the conditional Fatou lemma on $Y_n+Y-|X_n-X|.$
Fix $k \in \mathbb{N}.$ So for all $\epsilon>0, n \in \mathbb{N},$ $$P(E[|X_n-X||\mathcal{Q}]>\epsilon) \leq P(E[|X_n -\max(-k,\min(X_n,k))||\mathcal{Q}]>\epsilon/2)+P(E[|\max(-k,\min(k,X_n))-X||\mathcal{Q}]>\epsilon/2).$$ So, $$P(E[|X_n-X||\mathcal{Q}]>\epsilon) \leq P(E[Y_n - \min(Y_n,k)|\mathcal{Q}]>\epsilon/2)+\frac{2}{\epsilon}E[E[|\max(-k,\min(k,X_n))-X||\mathcal{Q}]]$$ Finally $$P(E[|X_n-X||\mathcal{Q}]>\epsilon) \leq \frac{2}{\epsilon}(E[Y_n] - E[\min(Y_n,k)]+\frac{2}{\epsilon}E[|\max(-k,\min(k,X_n))-X|]$$ Then taking $n \to \infty$ (applying dominated convergence theorem) $$\limsup_n P(E[|X_n-X||\mathcal{Q}]>\epsilon) \leq \frac{2}{\epsilon}(E[Y] - E[\min(Y,k)])+\frac{2}{\epsilon}E[|\max(-k,\min(k,X))-X|]$$ then taking $k \to \infty$ (dominated convergence theorem applied) $$\limsup_nP(E[|X_n-X||\mathcal{Q}]>\epsilon)$$
The first way seems faster (of course we need to extract 3 further subsequences from an arbitrary subsequence to use Fatou lemma)
We can also conclude a conditional version of Pratt lemma for both a.s convergence and convergence in probability.