Power-improved two-weight Bergman inequality

Let $p>1$, and let $\mu_1$ and $\mu_2$ be weights on the upper half-plane $\mathbb{R}_{+}^2$. For some $r>1$, consider the pair of powered weights $(\mu_1^r,\mu_2^r)$ in the two-weight class $A_p^+(\mathbb{R}_{+}^2)$, and use the associated inequality

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

Power-improved two-weight conjecture. If $(\mu_1^r,\mu_2^r)\in A_p^+(\mathbb{R}_{+}^2)$ for some $r>1$, then the displayed inequality holds for some $C>0$. This is presented as a variant motivated by a cited result, but the supplied source material gives no resolution; the conjecture therefore remains open.