Let $\mathbf{U}$ be a $n \times n$ Haar orthogonal matrix, $\mathbf{D}$ be a fixed diagonal matrix with half of its entries $+1$ and the remaining half $-1$ and $\left \langle\cdot, \cdot \right \rangle$ denote the Hilbert-Schmidt inner product. I would like to determine whether the following is true.
$$\max_{\left \| M\right \|_{HS}=1} \mathbb{E}\left \langle \mathbf{U}\mathbf{D}\mathbf{U}^*, \mathbf{M}\right \rangle^s = \mathbb{E}\left \langle \mathbf{U}\mathbf{D}\mathbf{U}^*, \frac{\mathbf{D}}{\sqrt{n}}\right \rangle^s, \: \forall s \geq 1$$
We note that it suffices to consider the statement for $s$ even, since $\left \langle \mathbf{U}\mathbf{D}\mathbf{U}^*, \mathbf{M}\right \rangle \overset{d}{\sim}- \left \langle \mathbf{U}\mathbf{D}\mathbf{U}^*, \mathbf{M}\right \rangle$, for all $\mathbf{M}$.
In addition, the statement is true for $s = 2$. This follows from expanding the power of the inner product and using results on fourth-order mixed moments of the entries of a Haar orthogonal matrix.
Any suggestion would be greatly appreciated!