Spectral eigenvalue conjecture for the Dirac soliton linearization

Let the spectral problem be the linearized spectral problem denoted by (third), with parameter $\omega\in(-1,1)$. Its isolated eigenvalues and end-point resonances are considered as functions of $\omega$. Spectral eigenvalue conjecture. The spectral problem has exactly two isolated eigenvalues and no end-point resonances for all $\omega\in(-1,1)$. The non-zero eigenvalue is positive for all $\omega\in(-1,0)$ and negative for all $\omega\in(0,1)$. This spectral assertion describes the eigenvalue configuration governing the stability analysis of the Dirac solitons. The supplied text does not indicate whether it has been proved or disproved.