Borcea's Toeplitz reformulation of the 2-variance conjecture

Assume $n\geq3$, choose $a_1,\dots,a_{n-1}\in\mathbb C$, and set

$$a_0=-\sum_{k=1}^{n-1}a_k.$$

Let $B$ be the $(n-1)\times(n-1)$ Toeplitz matrix

$$B=\begin{pmatrix}a_0&a_1&\dots&a_{n-2}\a_{n-1}&a_0&a_1&\dots\vdots&\ddots&\ddots&\vdots\a_2&\dots&a_{n-1}&a_0\end{pmatrix}.$$

Borcea's Toeplitz conjecture. The matrix $B$ has at least one eigenvalue $\lambda$ satisfying

$$|\lambda|^2\leq\sum_{k=1}^{n-1}|a_k|^2.$$

The source states that this matrix conjecture is equivalent to Statement (i), and hence to the 2-variance conjecture. No resolution is given.