Holomorphic stability characterization for weighted Bergman spaces

Let $\mathbb{D}$ be the unit disk and let $\mu$ be a boundary-accessable measure on $\mathbb{D}$, meaning that its support is not relatively compact in $\mathbb{D}$. Define

$$B^2(\mathbb{D},\mu)=L^2(\mathbb{D},\mu)\cap\mathcal{O}_h(\mathbb{D}),$$

where $\mathcal{O}_h(\mathbb{D})$ is the space of harmonic functions on $\mathbb{D}$, and let $h^1(\mathbb{D})$ be the harmonic Hardy space. A measure $\mu$ is a $(1,2)$-Carleson measure if there exists $C>0$ such that

$$\left(\int_{\mathbb{D}}|f(z)|^2\,\mu(dz)\right)^{1/2}\leq C\|f\|_{H^1(\mathbb{D})},\qquad f\in H^1(\mathbb{D}).$$

Holomorphic stability conjecture. For any boundary-accessable measure $\mu$ on $\mathbb{D}$, the pair $(B^2(\mathbb{D},\mu),h^1(\mathbb{D}))$ is holomorphically stable if and only if $\mu$ is a $(1,2)$-Carleson measure. The conjecture extends the theorem proved in the paper for radial boundary-accessable measures. The non-radial case is left open.