The Jordan-curve conjecture for banded Toeplitz symbols

Let $f_m$ be the symbol function defined by the paper, and let $f_m^{-1}(\lambda)$ denote the preimage of the symbol function, namely the roots of $f_m(z)-\lambda=0$. Let $\mathbf{T}_n(f_m)$ be the corresponding $n\times n$ banded Toeplitz matrix, and let $\sigma(\mathbf{T}_n(f_m))$ denote its spectrum.

Jordan-curve conjecture. The following statements are equivalent:

1. There exists a Jordan curve parametrised by the complex band structure.
2. $\displaystyle \lim_{n\to\infty}\sigma\bigl(\mathbf{T}_n(f_m)\bigr)\subset\mathbb{R}$.

The conjecture is motivated by an earlier result that was presented as a theorem under weaker assumptions, but whose proof was later found to be erroneous. The equivalence is therefore stated as a conjecture and remains open according to the supplied status information.