Let $\mathcal{F}$ be the space of all functions that uniformly and independently map the alphabet $\mathcal{X}$ to the set $\{1,2,\ldots,A\}$. Let $p(x|y)$ be an arbitrary conditional probability distribution. Does the following inequality hold? \begin{align} \mathbb{E}_\mathcal{F}\left[\sqrt{\sum_{x\in\mathcal{X}}p^2(x|y)\mathbf{1}[\mathcal{F}(x)=1]}\right]&\overset{?}{\le}\sqrt{\sum_{x\in\mathcal{X}}p^2(x|y)}\mathbb{E}_\mathcal{F}\left[\mathbf{1}[\mathcal{F}(x)=1]\right]\\ &=\frac{1}{A}\sqrt{\sum_{x\in\mathcal{X}}p^2(x|y)}, \end{align} where $\mathbf{1}[\cdot]$ is the indicator function.
I know that using Jensen's inequality for $\sqrt{x}$ we have: \begin{align} \mathbb{E}_\mathcal{F}\left[\sqrt{\sum_{x\in\mathcal{X}}p^2(x|y)\mathbf{1}[\mathcal{F}(x)=1]}\right]&\leq\sqrt{\sum_{x\in\mathcal{X}}p^2(x|y)\mathbb{E}_\mathcal{F}\left[\mathbf{1}[\mathcal{F}(x)=1]\right]}\\ &=\frac{1}{\sqrt{A}}\sqrt{\sum_{x\in\mathcal{X}}p^2(x|y)}. \end{align}