Let $X=\begin{pmatrix} r & x\\ \vert y\vert e^{i\theta} & r\end{pmatrix}$ where $r,x>0, y\in\mathbb{C}$ and $\theta\in [0,2\pi)$ and $Y=\begin{pmatrix} r & x\\ \vert y\vert & r\end{pmatrix}$. Is $w(X)=w(Y)$ where $w(X):=\sup\limits_{\Vert\alpha\Vert=1}\vert\langle X\alpha,\alpha\rangle\vert$ is called numerical radius of $X$?
Comments: I observed that it is true whenever $r=0$. Do you think $X$ and $Y$ are unitarily equivalent which will give the affirmative conclusion of the question?