Essentially Hermitian conjecture for block-matrix norm inequalities

Let $X\in\mathbb{M}_n$. Say that $X$ has the universal block-matrix inequality if, for every positive block matrix with $X$ as its off-diagonal block,

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

Essentially Hermitian conjecture. If $X$ has the universal block-matrix inequality, then $X$ is essentially Hermitian. The conjecture is motivated by a theorem proving the converse implication under the additional hypothesis that $X$ is invertible with distinct singular values; the supplied text does not report a resolution of the general claim.