Determinant of "quasi-circulant" matrices
Here is a counter example:
$$ \left[\begin{matrix}0 & 0 & -1.0 & 0 & -1.0 & -1.0\\-1.0 & 0.391134657778849 & 0 & -1.0 & 0 & -1.0\\-1.0 & -1.0 & 0.391134657778849 & 0 & -1.0 & 0\\0 & -1.0 & -1.0 & 0.391134657778849 & 0 & -1.0\\-1.0 & 0 & -1.0 & -1.0 & 0.391134657778849 & 0\\0 & -1.0 & 0 & -1.0 & -1.0 & 0.391134657778849\end{matrix}\right] $$
It has a determinant of 10.609672399765413.
Here is another one (determinant of 0.99): $$ \left[\begin{matrix}0 & 0 & -1.0 & 0\\0 & 0.10 & 0 & -1.0\\-1.0 & 0 & 0.10 & 0\\0 & -1.0 & 0 & 0.10\end{matrix}\right] $$