Two-weight Bergman inequality without the pointwise weight comparison condition

Consider the upper half-plane $\mathbb{R}_{+}^2=\{z\in\mathbb{C}:\operatorname{Im}z>0\}$, two weights $\mu_1$ and $\mu_2$ on $\mathbb{R}_{+}^2$, and the two-weight inequality

$$\int_{\mathbb{R}_+^2}\left|\int_{\mathbb{R}_+^2}-\frac{f(w)}{(z-\overline{w})^2}\,dA(w)\right|^p\mu_1(z)\,dA(z)\le C\int_{\mathbb{R}_+^2}|f(z)|^p\mu_2(z)\,dA(z).$$

Here $(\mu_1,\mu_2)\in A_p^+(\mathbb{R}_+^2)$ denotes the relevant two-weight Muckenhoupt condition. The two-weight Bergman inequality conjecture. For $p>1$, if $\mu_1$ and $\mu_2$ satisfy $(\mu_1,\mu_2)\in A_p^+(\mathbb{R}_{+}^2)$, then the displayed inequality holds for some $C>0$.

This asks whether the pointwise hypothesis $c\mu_1\geq\mu_2$ used in the proposition can be removed from the sufficient direction. It is presented as an open problem related to the two-weight inequality.