Extension of the block-matrix majorization inequality to normal off-diagonal blocks

Let a positive semidefinite matrix in $\mathbb{M}_{2n}^{+}$ be written in blocks of the same size, with normal off-diagonal blocks. Normal-block majorization conjecture. The inequality from Corollary 1.3 should still hold: for every symmetric norm,

$$\left\|\begin{bmatrix} A & X \\ X & B\end{bmatrix}\right\|\leq \|A+B\|.$$

The preceding result establishes this inequality when the off-diagonal blocks are Hermitian; the conjecture asks whether normality suffices. The paper gives no resolution, so the question remains open.