Expected value property concerning two random variables Given two bounded real-valued random variables $X, Y$, is it necessarily the case that
$$\mathbb{E}_{x,y \sim (X, Y)} [(x + y)^2] \ge \mathbb{E}_{x, y \sim (X, Y)} [(\mathbb{E}[X \mid y] + \mathbb{E}[Y \mid x])^2]?$$
 A: (Solution courtesy Holden Lee.)
We have
\begin{align*}
\mathbb{E}[((X - \mathbb{E}[X \mid Y]) - (Y - \mathbb{E}[Y \mid X]))^2] &\ge 0\\
\mathbb{E}[(X - \mathbb{E}[X \mid Y])^2 + (Y - \mathbb{E}[Y \mid X])^2] &\ge 2 \mathbb{E}[(X - \mathbb{E}[X \mid Y])(Y - \mathbb{E}[Y \mid X])].
\end{align*}
Now, observe that
$$\mathbb{E}[(X - \mathbb{E}[X \mid Y])^2 = \mathbb{E}[X^2] + \mathbb{E}[\mathbb{E}[X \mid Y]^2] - 2\mathbb{E}[X \mathbb{E}[X \mid Y]] = \mathbb{E}[X^2] - \mathbb{E}[\mathbb{E}[X \mid Y]^2],$$
where in the last step we use the fact that
\begin{align*}
\mathbb{E}[X \mathbb{E}[X \mid Y]] &= \mathbb{E}_{y \sim Y}[\mathbb{E}_{x \sim X \mid Y = y}[x \mathbb{E}[X \mid Y = y]]]\\
&= \mathbb{E}_y[\mathbb{E}[X \mid Y = y] \mathbb{E}_{x \sim X \mid Y = y}[x]] = \mathbb{E}[\mathbb{E}[X \mid Y]^2].
\end{align*}
Similarly we have that $\mathbb{E}[(Y - \mathbb{E}[Y \mid X])^2] = \mathbb{E}[Y^2] - \mathbb{E}[\mathbb{E}[Y \mid X]^2]$. Additionally we have that
$$\mathbb{E}[X \mathbb{E}[Y \mid X]] = \mathbb{E}_x[x \mathbb{E}[Y \mid X = x]] = \mathbb{E}_x[\mathbb{E}[XY \mid X = x]] = \mathbb{E}[XY].$$
Together with these facts, we adapt out earlier equation:
$$\mathbb{E}[X^2] - \mathbb{E}[\mathbb{E}[X \mid Y]^2] + \mathbb{E}[Y^2] - \mathbb{E}[\mathbb{E}[Y \mid X]^2] \ge 2 \mathbb{E}[\mathbb{E}[X \mid Y] \mathbb{E}[Y \mid X]] - 2 \mathbb{E}[XY].$$
Rearranging terms:
\begin{align*}
\mathbb{E}[X^2 + 2XY + Y^2] &\ge \mathbb{E}[\mathbb{E}[X \mid Y]^2] + 2 \mathbb{E}[X \mid Y] \mathbb{E}[Y \mid X] + \mathbb{E}[\mathbb{E}[Y \mid X]^2]\\
\mathbb{E}[(X + Y)^2] &\ge \mathbb{E}[(\mathbb{E}[X \mid Y] + \mathbb{E}[Y \mid X])^2],
\end{align*}
as desired.
