Absence of pure point spectrum under dynamical avoidance

Let $d_\infty$ and $N$ be the parameters in the spectral construction, let $\phi$ be the associated map, let $g$ be the associated rational map, and let $I_{g^n}$ denote the indeterminacy points of $g^n$. For every real $\lambda$, assume there are open subsets $U\subset\mathbb C$ and $V\subset\mathbb L^G$ such that

$$\lambda\in U,\qquad \bigcup_{n=1}^{\infty} I_{g^n}\subset V,\qquad g^n(\phi(U))\cap V=\emptyset.$$

Pure point spectrum conjecture. If $d_\infty=N$ and this condition holds for every real $\lambda$, then

$$\Sigma_{pp}=\emptyset.$$

The question arises because the non-deterministic spectrum is related to the indeterminacy points of $g$, while pure point spectrum beyond the non-deterministic component would suggest that iterates of $g$ approach those points. The supplied text gives no resolution, so the conjecture remains open.