Let $B\in M_n(\mathbb{C})$ with $BB^*+B^*B=I$. Is $ \begin{pmatrix} 0 & B \\ B & 0 \\ \end{pmatrix} $ is unitarily similar to $ \begin{pmatrix} 0 & \vert B\vert \\ \vert B^*\vert & 0 \\ \end{pmatrix} $ where $\vert A\vert=\sqrt{A^*A}$ for $A\in M_n(\mathbb{C})$?
Comments: I have been trying to prove the above question in the affirmative using the polar decomposition of $B$ but could not be successful yet. I also think $BB^*+B^*B=I$ may not be needed to prove the above question in the affirmative. Nevertheless, I include the condition in the question since I have it beforehand.
Any comment is highly appreciated. Thanks in advance.
As noted in the comment on your question, the statement you are interested in is not generally true. However, you might be interested in the fact that the matrices $$ \pmatrix{0 & B\\ B^* & 0}, \quad \pmatrix{0&|B|\\ |B| & 0} $$ are necessarily unitarily similar. We note that $B$ has a polar decomposition of the form $$ B = |B|U $$ for some unitary matrix $U$. With that, we can write $$ \pmatrix{0&B\\B^*&0} = \pmatrix{0 & |B| U\\ U^*|B| & 0} =\\ \pmatrix{I & 0\\0 & U}^*\pmatrix{0 & |B|\\ |B| & 0}\pmatrix{I & 0\\0 & U}. $$