Operator-theoretic Borel-series zero conjecture

Let $\{z_n\}_{n\in\mathbb{N}}$ be distinct points in the open unit disk converging to $1$. Let $\mathcal{H}$ be a separable infinite-dimensional complex Hilbert space with orthonormal basis $\{\mathbf e_n\}_{n\in\mathbb{N}}$, and let $A$ be the bounded normal operator defined by $A\mathbf e_n=z_n\mathbf e_n$. If $P$ is a bounded operator on $\mathcal{H}$ such that $I-P$ is a rank-one orthoprojection, then the compression $PAP|_{P\mathcal{H}}$ has infinite point spectrum. Operator-theoretic Borel-series zero conjecture. Under these hypotheses, the stated compression has infinitely many eigenvalues. The source says this is equivalent to the preceding Borel-series conjecture after normalization, while a later remark discusses possible relaxations; no resolution is given.