Consider the following $n\times n$ real-valued matrix:

$$ A=\begin{bmatrix} \alpha_1 & \beta_1 & 0 &\cdots & 0 &\gamma_n\\ \gamma_1 & \alpha_2 & \beta_2 & \cdots & \cdots & 0\\ 0 & \ddots & \ddots & \ddots & \cdots & \vdots\\ \vdots & \cdots & \ddots & \ddots & \ddots & 0\\ 0 & \cdots & \cdots & \gamma_{n-2} & \alpha_{n-1} & \beta_{n-1}\\ \beta_n & 0 & \cdots & \cdots & \gamma_{n-1} & \alpha_n \end{bmatrix} $$

There exists a triangular matrix $T$ s.t. $N=T^{-1}AT$ is a normal matrix (that is $N N^\top= N^\top N$)? If so, what is the form of $T$ in terms of $\alpha_i$, $\beta_i$ and $\gamma_i$?

Any help (and/or links to papers which address this or related problems) will be appreciated.

**EDIT.**
The answer is false, if the choice of the coefficients $\alpha_i$, $\beta_i$ and $\gamma_i$ is arbitrary. What can we say if we make the additional assumption that $\alpha_i\neq 0$, $\beta_i\neq 0$ and $\gamma_i\neq 0$ for all $i$?