Brevig–Ortega-Cerdà–Seip–Zhao contractive inequality for weighted Bergman spaces

Let $H^2$ be the Hardy space on the unit disk, let $A^p_\beta$ denote the standard weighted Bergman space with exponent $p$ and weight parameter $\beta$, and let the Szegő kernel be $K_\zeta(z)=(1-\overline{\zeta}z)^{-1}$ for $\zeta\in\mathbb{D}$. Brevig–Ortega-Cerdà–Seip–Zhao conjecture. If $p>2$, then

$$\|f\|_{A^p_{\frac{p}{2}-2}}\leq\|f\|_{H^2},\qquad \forall f\in H^2,$$

and equality is attained if and only if $f$ is a constant multiple of the Szegő kernel. This extends Carleman's inequality and Burbea's theorem from integer parameters to all real $p>2$; the source describes substantial evidence in its favour, but the conjecture remains open in the supplied text.