Let $X_n = (X_1^n, X_2^n) \to (X_1, X_2)$ in distribution ($\mathbb{R}^2$-valued) and suppose that $(X^n)$ is uniformly integrable, i. e. $$ \lim_{a \to \infty} \sup_n E[1_{\|X_n\| \geq a} \|X_n\|] = 0. $$ Further, assume continuous convergence of the conditional distributions, i.e. $$ \mathcal{L}(X_1^n \mid X_2^n = x_2^n) \to \mathcal{L}(X_1 \mid X_2 = x_2) $$ weakly for each converging $x_2^n \to x_2$.
$\textbf{ Question: }$ Is $\mathcal{L}(X_1^n \mid X_2^n = x_2^n)$ uniformly integrable, i.e. do we have $$ \lim_{a \to \infty} \sup_n E[1_{|X_1^n| \geq a} |X_1^n| \mid X_2^n = x_2^n] = 0 $$ for each $x_2^n \to x_2$ ? (This notation refers to a sequence of regular versions of the conditional distributions).
$\textbf{ Approach: } $ It can be shown that $E[X_1^n \mid X_2^n]$ is u.i. However, I don't see how this connects to the above. Thanks!