Reality conjecture for the limiting spectrum of real block Toeplitz matrices

Let $B(z)$ be a real $k\times k$ matrix-valued symbol. Define

$$\Gamma(B)=\{z\in\mathbb{C}:\exists\lambda\in\mathbb{R}\text{ such that }f(z,\lambda)=0\},$$

where $f(z,\lambda)$ is the characteristic function of $B$, and let $\Lambda_0(B)$ denote the set of limiting spectral points associated with the $q$-th and $(q+1)$-st roots of $z^qf(z,\lambda)=0$ having equal modulus. Reality conjecture. The set $\Lambda_0(B)$ belongs to $\mathbb{R}$ if and only if $\Gamma(B)$ contains $k$ Jordan curves (ovals) having $0$ in their interior. This conjecture generalizes the scalar Laurent-polynomial case, where $\Gamma(B)$ is the inverse image of $\mathbb{R}$ under the symbol and the corresponding curve has been studied previously. The statement is presented as a general conjecture for real banded block Toeplitz matrices; no resolution is supplied in the source.