Finiteness criterion for non-real Dirac eigenvalues near the thresholds

Let $D_m(b,V)$ be the Dirac operator from Assumption 1.1, with $V=\Phi W$, where $W$ is Hermitian of definite sign and ${\rm Arg}\,\Phi\in\mathbb{C}\setminus\mathbb{R}e^{ik\frac{\pi}{2}}$ for some $k\in\mathbb{Z}$. For a domain $\Omega$ as in the definition of $\mathcal{N}_{\pm m}$, let $\mathcal{N}_{\pm m}(D_m(b,V),\Omega)$ denote the corresponding count of non-real eigenvalues near the thresholds $\pm m$. Finiteness criterion. One has

$$\mathcal{N}_{\pm m}\bigl(D_m(b,V),\Omega\bigr)<\infty$$

if and only if $\pm\operatorname{Re}(V)>0$. The conjecture proposes a general criterion governing whether non-real eigenvalues accumulate near the spectral thresholds $\pm m$; the supplied context does not state a proof or a resolution, so its status remains open.