On a conjecture about contractive inequalities for weighted Bergman spaces

We are interested in the norm of the inclusion between the standard weighted Bergman spaces $A^2_\alpha$ and $A^p_{\frac{p}{2} (\alpha + 2)-2}$, $p > 2$, which is conjectured to be contractive by O.F. Brevig, J. Ortega-Cerd\`a, K. Seip and J. Zhao (in the limit case $\alpha = -1$) and E. Lieb and J.P. Solovej (for $\alpha > -1$). We provide some partial answers and improvements of some recent results.