Bound the conditional expectation of a random matrix under weak dependence

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 $$$$\label{eq:main} \mathbb{E}\big[\|\mathbb{E}[X|\mathcal{G}]\|_2\big]\leq Cd\phi,$$$$ where $$C$$ is some constant and $$d$$ is the dimension.

From Lemma 4.4.1 of this paper, I already know that $$$$\mathbb{E}\big[|\mathbb{E}[X|\mathcal{G}]|\big]\leq C\phi,$$$$ hold if $$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.

By elementary inequality of matrix, we know that $$\begin{equation*} |X|_{\infty}\leq\|X\|\leq 1, \quad\text{ and }\quad\|X\|\leq \|X\|_F. \end{equation*}$$ For every element $$X_{i,j}$$ of $$X$$, by Lemma 4.4.1 of this paper we have that $$\begin{equation*} \mathbb{E}\Big[\big|\mathbb{E}[X_{ij}|\mathcal{G}]\big|^2\Big]\leq \mathbb{E}\Big[\big|\mathbb{E}[X_{ij}|\mathcal{G}]\big|\Big]\leq 2\pi\phi. \end{equation*}$$ Therefore, we have that \begin{align*} &\mathbb{E}\Big[\|\mathbb{E}[X|\mathcal{G}]\|\Big]\leq \mathbb{E}\Big[\|\mathbb{E}[X|\mathcal{G}]\|_F\Big],\\ \leq&\Big\{\mathbb{E}\Big[\|\mathbb{E}[X|\mathcal{G}]\|^2_F\Big]\Big\}^{1/2}\\ \leq&\Big\{\mathbb{E}\Big[\sum_{i,j=1}^d\big|\mathbb{E}[X_{ij}|\mathcal{G}]\big|^2\Big]\Big\}^{1/2}\leq\sqrt{d^22\pi\phi}=d\sqrt{2\pi\phi}. \end{align*}