Abstract Birkhoff-regularity conjecture for completely continuous operators

Let $G$ be an abstract completely continuous operator, let $\Lambda:=\left\{\lambda_j\right\}$ be the set of its eigenvalues, and fix $\delta>0$. For each $\lambda$, define the circle

$$K_\lambda:=\left\{\left|z-\lambda\right|\leq\delta\cdot\left(1+\left|\operatorname{Im}\lambda\right|\right)\right\}.$$

Let $T_\varepsilon^\pm$ be the sectors referred to in the paper. Abstract Birkhoff-regularity conjecture. Existence of an invertible limit of the characteristic function of $G$ in the sectors $T_\varepsilon^\pm$, away from the circles $K_{\lambda_j}$ and $K_{\overline{\lambda_j}}$ for $\lambda_j\in\Lambda$, is a correct reformulation of the Birkhoff-regularity condition and yields unconditional basicity of the eigenvectors of $G$, perhaps with brackets. This proposed abstract reformulation is intended to connect characteristic-function asymptotics with Birkhoff regularity and spectral decompositions. The source presents it as a supposition and does not provide a proof or a resolution.