A characterization of essentially Hermitian off-diagonal blocks

For $X\in\mathbb{C}^{n\times n}$ the following are equivalent: (1) $\|P\|\le\|A+B\|$ in the operator norm for every positive semidefinite completion $P=\begin{pmatrix}A&X\\X^*&B\end{pmatrix}$; (2) the same holds whenever $A+B$ is a scalar multiple of $I$; (3) for every Hermitian $T$ the largest and smallest eigenvalues of $\begin{pmatrix}T&X\\X^*&-T\end{pmatrix}$ are negatives of each other; (4) $X=\alpha H+\beta I$ with $H$ Hermitian and $\alpha,\beta\in\mathbb{C}$. No invertibility or distinct-singular-value hypothesis is needed. Completions with scalar diagonal sum give (2)$\Rightarrow$(3). Second-order eigenvalue perturbation then yields $|u^*Xv|=|v^*Xu|$ for all orthonormal pairs, which makes $X$ normal, and a test with cube roots of unity makes every three eigenvalues collinear. Conversely, a unitary change of basis splits the matrix in (3) into $2\times2$ blocks with symmetric spectra.