I have figured out to prove a relatively weaker result.

 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][1] 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*}

  [1]: https://projecteuclid.org/journals/annals-of-probability/volume-7/issue-1/Approximation-Thorems-for-Independent-and-Weakly-Dependent-Random-Vectors/10.1214/aop/1176995146.full