Validity of the Jordan-curve criterion for a real Toeplitz spectrum

Determine whether the reality of the open spectrum of a banded non-Hermitian Toeplitz matrix is characterized by the existence of a Jordan curve in the real-level set of its symbol that winds around the origin, without assuming that the generalized Brillouin zone is parametrized by a $C^1$ polar curve.

Background

For a banded non-Hermitian Toeplitz symbol, the paper discusses an earlier proposed criterion asserting that the open limiting spectrum is real under a geometric condition on the real-level set of the symbol: the set $f_m^{-1}(mathbb{R})$ should contain a Jordan curve winding around the origin. The criterion would be more general than the sufficient condition established in the paper, which assumes that the generalized Brillouin zone is parametrized by a $C^1$ polar curve.

The cited proof of the earlier criterion was withdrawn, and the paper reports that counterexamples had been announced without the criterion having been restored. Under the paper's strict middle-root separation assumption, the authors instead prove that a real open spectrum is equivalent to the generalized Brillouin zone being spanned by an analytic polar curve. Thus, the broader Jordan-curve criterion remains an explicitly identified conjectural or unresolved issue outside the assumptions and characterization proved in the paper.