Joint Continuity of Conditional Expectation under Mean-Square Convergence

Let $(\Omega,\mathcal{F},P)$ be a probability space. Let $(X_n)_{n\in\mathbb{N}}$ be a sequence of square-integrable random variables and $X$ a square-integrable random variable on $(\Omega,\mathcal{F},P)$ such that the mean-square distances $(\lVert X_n-X\rVert_{2})_{n\in\mathbb{N}}$ have limit $0$. Let $(\mathcal{G}_n)_{n\in\mathbb{N}}$ be a sequence of sub-$\sigma$-algebras of $\mathcal{F}$ that converges in mean square to a sub-$\sigma$-algebra $\mathcal{G}$ of $\mathcal{F}$.

Then for every choice of conditional expectations $Y_n$ of $X_n$ given $\mathcal{G}_n$ ($n\in\mathbb{N}$) and $Y$ of $X$ given $\mathcal{G}$, the real sequence $(\lVert Y_n-Y\rVert_{2})_{n\in\mathbb{N}}$ has limit $0$. In the notation of Conditional Expectation of a Square-Integrable Random Variable: