Transfering order to conditional expectation

Question

Suppose that $Q\sim P$, $X,Y\in L^2(\mathcal{F})$ are such that $$ \mathbb{E}_{Q}[(X-Y)^2] \leq \mathbb{E}_{P}[(X-Y)^2]. $$ Under what circumstances, can we conclude that $$ \mathbb{E}_{Q}[( \mathbb{E}_{Q}[X|\mathcal{G}]- Y)^2] \leq \mathbb{E}_{P}[( \mathbb{E}_{P}[X|\mathcal{G}] -Y)^2]? $$ Where $\mathcal{G}$ is a fixed sub $\sigma$-algebra of $\mathcal{F}$.

Answer 1

A straightforward calculation shows that the second inequality holds if, in addition, \begin{align} &\mathsf{E}_P\operatorname{Var}_P(X\mid \mathcal{G})-\mathsf{E}_Q\operatorname{Var}_Q(X\mid \mathcal{G})\\ &\qquad\le 2\left(\mathsf{E}_P\operatorname{Cov}_P(X,Y\mid \mathcal{G})-\mathsf{E}_Q\operatorname{Cov}_Q(X,Y\mid \mathcal{G})\right). \end{align}