Core-carrier conjecture for analytic weighted Bergman spaces

Let $G$colon $(0,\infty)\to(0,\infty)$ be the function occurring in the measure structure defined by, let $w\in\mathcal{L}^1(\mathbb{T})$ be a non-negative weight, and let

$$\int_0^d \log(1/G(x))\,dx=\infty$$

for every $d>0$. Write $\operatorname{core}(w)$ for the core of $w$, and let $\mathcal{P}^2(\mu)$ be the space associated with a measure $\mu$ of the form. Core-carrier conjecture. If $\mathcal{P}^2(\mu)$ is a space of analytic functions on $\mathbb{D}$, then $\operatorname{core}(w)$ is a carrier for $w$. This would give a sharp version of the irreducibility result for the spaces under consideration, replacing the exponential-decay hypothesis by the non-integrability of $\log G$; the source does not state whether the conjecture is resolved.