Real-valued Jordan curves and nonreal limiting spectra of banded Toeplitz matrices

We construct Laurent polynomials that are real-valued on a Jordan curve surrounding the origin but whose banded Toeplitz matrices have nonreal limiting spectrum. This refutes the weakest surviving form of a conjecture of Shapiro and Štampach, and consequently also the implication to real spectra of all finite sections. The main example $b=h^2$, with $h(z)=z^2+97z+103-10201/z$, has real coefficients and takes values in $[-4000000,0]$ on a real-analytic Jordan curve of winding number one about $0$, obtained with Rouché's theorem as the set of $z$ with $|z|<30$ and $h(z)\in\mathrm i[-2000,2000]$. Near $\lambda_0=(20192/97)^2$, the Schmidt–Spitzer equal-modulus characterisation and an implicit-function argument show that the limiting spectrum is a regular real-analytic arc crossing the real axis transversely. A second symbol, $a(z)=10201/z+97z+\mathrm{i}(103-z^2)$, has the explicit nonreal limiting spectral point $(20192/97)\mathrm{i}$.