Lieb–Solovej generalized contractive inequality for weighted Bergman spaces

Let $\alpha>-1$, let $A^2_\alpha$ be the standard weighted Bergman space, and let $A^p_\beta$ denote the corresponding standard weighted Bergman space with exponent $p$ and weight parameter $\beta$. The Bergman kernel of $A^2_\alpha$ is

$$K_\zeta(z)=(1-\overline{\zeta}z)^{-(\alpha+2)},\qquad \zeta\in\mathbb{D}.$$

Lieb–Solovej conjecture. If $p>2$, then

$$\|f\|_{A^p_{\frac{p}{2}(\alpha+2)-2}}\leq\|f\|_{A^2_\alpha},\qquad \forall f\in A^2_\alpha,$$

and equality is possible if and only if $f$ is a constant multiple of the Bergman kernel. This generalizes the Brevig–Ortega-Cerdà–Seip–Zhao question from the limiting Hardy-space case to all $\alpha>-1$; the supplied text presents it as a question stated by Lieb and Solovej, with no resolution given.