Operator-theoretic Clunie–Eremenko–Rossi conjecture

Let $\{\alpha_n\}_{n\in\mathbb{N}}$ be unimodular complex numbers and let $\{\beta_n\}_{n\in\mathbb{N}}$ be positive numbers converging to $0$, with the ratios $\alpha_n\beta_n^{-1}$ distinct. Define operators on a Hilbert space $\mathcal{H}$ with orthonormal basis $\{\mathbf e_n\}$ by

$$A\mathbf e_n=\alpha_n\mathbf e_n,\qquad B\mathbf e_n=\beta_n\mathbf e_n.$$

Thus $A$ is unitary and $B$ is compact selfadjoint. For a totally non-zero unit vector $\mathbf v$, let $P$ be the orthoprojection onto $\mathcal{K}=\mathbf v^\perp$, and set $A'=PAP|_{\mathcal{K}}$ and $B'=PBP|_{\mathcal{K}}$. Operator-theoretic Clunie–Eremenko–Rossi conjecture. The $B'$-point spectrum of $A'$ is infinite:

$$|\Sigma_{B'}(A')|=\infty,$$

that is, there are infinitely many $z\in\mathbb{C}$ such that $N(A'-zB')\ne\{0\}$. This is presented as equivalent to the classical Clunie–Eremenko–Rossi conjecture; the supplied text gives no resolution.