= Solution
Take $X_n=Y_nZ_n$ and $\mathcal G=\sigma(Y_1,Y_2,\ldots)$. Then $\mathbb P(X_n=n)=n^{-2}$, so $(X_n)$ is uniformly integrable and $X_n\to0$ almost surely by the <Borel-Cantelli lemmas>. Independence and $\mathbb EZ_n=1$ give
$$
\mathbb E[X_n\mid\mathcal G]=Y_n.
$$
But the independent events $\{Y_n=1\}$ have divergent probability sum, so the second Borel-Cantelli lemma makes them occur infinitely often. Thus these conditional expectations do not converge almost surely to zero.
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