Spectral instability conjecture for stationary states on the tadpole graph

Let $(u_{n,\omega}^{\pm},0)$ be the stationary standing-wave branch indexed by $n\in\mathbb{N}$, and let the spectral stability problem be the linearized eigenvalue problem for the nonlinear Schrödinger equation on the tadpole graph. Write the frequency as

$$\omega=-\epsilon^2,$$

where $\epsilon>0$ is sufficiently small. Spectral instability conjecture. For every $n\in\mathbb{N}$, the branch $(u,v)=(u_{n,\omega}^{\pm},0)$ is spectrally unstable, with at least $n$ quartets of complex eigenvalues $\lambda$ in the spectral stability problem. The preceding eigenvalue count is inconclusive for these stationary states, so the conjecture concerns their instability for small negative $\omega$; the supplied text gives no resolution.