In their paper, "Corners of Normal Matrices," R. Bhatia and M.D. Choi ask the following question: Given a matrix pair $(B,C)$ where $B,C∈M_n$, does there exist matrices $A,D ∈ M_n$ such that the block matrix:

$N = \begin{bmatrix} A & B \\ C & D \end{bmatrix}$

is normal. I have made some progress on this problem by constructing explicit normal matrices of the form above for certain pairs of matrices $(B,C)$ that do not appear in the literature. One open question in the paper is whether or not the quantity $α_n=\sup\{\frac{||B||}{||C||}:∃A,D ∈ M_n$ such that N is normal $\}$ is bounded. Moreover, they prove that $α_n < \sqrt{n}$ for $n > 4$. I suspect that $α_n$ is unbounded but I can only get that $α_n \geq \sqrt{2}$. Does anyone have any ideas here?

**Edit after setting the bounty**

What I am looking for, if it exists, is a sequence of $n \times n$ matrices, $A_n, B_n, C_n, D_n$ such that when you put them in the matrix configuration above, the resulting matrix is normal and $\frac{||B_n||}{||C_n||} \to \infty$ as $n \to \infty$. Alternatively, if you can prove that my intuition is wrong, i.e. show that $\alpha_n$ is bounded, then you will also receive the bounty.

**Second Edit**
The norm we are considering here is the usual operator norm.