Jordan-curve conjecture for eigenvalue contour lines of Kac–Murdock–Szegő matrices

Let $L_{n,N}^{(1)}$ and $L_{n,N}^{(2)}$ denote the two types of closed eigenvalue contour lines associated with the Kac–Murdock–Szegő matrix of size $n$, and let $N_{\mathrm{min}}(n)$ be the lower threshold for the parameter $N$. Jordan-curve conjecture. The closed curves $L_{n,N}^{(1)}$ and $L_{n,N}^{(2)}$ are Jordan curves whenever $N\ge n$, but are non-Jordan whenever $N_{\mathrm{min}}(n)<N<n$. This conjecture summarizes the observed transition from loops below the borderline value $N=n$ to Jordan curves at and above it, extending a previously stated conjecture to the case $N\ne n$; the general assertion remains open.