Cyclicity conjecture for singular inner functions in analytic weighted spaces

Let $G$ and $w\in\mathcal{L}^1(\mathbb{T})$ be as in the core-carrier conjecture, let $\mu$ have the form, and let $\operatorname{core}(w)$ denote the core of $w$. For a singular measure $\nu$, let $S_\nu$ be the associated singular inner function. Cyclicity conjecture. In this setting, $S_\nu$ is cyclic in the space of analytic functions $\mathcal{P}^2(\mu)$ if and only if

$$\nu\bigl(\operatorname{core}(w)\bigr)=0.$$

This is proposed as a strong version combining the cited cyclicity result with the paper's cyclicity theorem; the source gives no evidence that it has been resolved.