Let $X$ be an $d\times d$ random matrix satisfying $\mathbb{E}[X]=0$ and $\|X\|_2\leq 1$ almost everywhere. Let $\mathcal{F}$ be the $\sigma$-field generated by $X$. Now suppose we have another $\sigma$-field $\mathcal{G}$, it satisfies that \begin{equation*} \rho(\mathcal{F},\mathcal{G})=\sup_{A\in\mathcal{F},B\in\mathcal{G}}|\mathbb{P}(AB)-\mathbb{P}(A)\mathbb{P}(B)|\leq\phi. \end{equation*} Now I want to prove that \begin{equation}\label{eq:main} \mathbb{E}\big[\|\mathbb{E}[X|\mathcal{G}]\|_2\big]\leq Cd\phi, \end{equation} where $C$ is some constant and $d$ is the dimension.

From Lemma 4.4.1 of this paper, I already know that \begin{equation} \mathbb{E}\big[|\mathbb{E}[X|\mathcal{G}]|\big]\leq C\phi, \end{equation} hold if $X$ is a scalar random variable. Now I want to extend this result to matrix case.

I tried to use the discretization technique as in Proposition 5.17 of Wainwright's book, but then I can only prove it bounded by $C9^d\phi$, which is undesirable because it is exponentially related to the dimension $d$. So I hope someone can give me some idea about it.