Correlation of conditional expectation of uncorrelated random variables

Question

Let $X,Y\in\mathcal{L}_{2}\left(\Omega,\mathcal{F},\mathbb{P}\right)$ satisfy $\mathbb{E}\left[X\right]=\mathbb{E}\left[Y\right]=\mathbb{E}\left[XY\right]=0$, and $\mathbb{E}\left[X^2\right]=\mathbb{E}\left[Y^2\right]=1$. The problem asks to show (i) $\mathbb{E}\left[\mathbb{E}\left[X|\mathcal{G}\right]\mathbb{E}\left[Y|\mathcal{G}\right]\right]\leq\frac12$ for any sub $\sigma$-algebra $\mathcal{G}\subset \mathcal{F}$, and (ii) to find $\mathcal{G}\subset\mathcal{F}$ for which the equality holds, assuming that $X,Y$ are i.i.d. Any help is appreciated. Thanks in advance!