The conjecture on the sharp norm of the Bergman projection

Let $n>1$ and $\beta$ be the (semi) norm for which the Bergman projection $P_\alpha$ has norm $\|P_\alpha\|_{\tilde\beta}=\tilde C_{\alpha,n}$. With $C_{\alpha,n}$ as in Theorem Mait, one has the estimate

$$\frac{\pi}{2}C_{\alpha,n}\leq \tilde C_{\alpha,n}\leq \frac{\sqrt{\pi^2+4}}{2}C_{\alpha,n}.$$

Sharp Bergman projection norm conjecture. The upper quantity is expected to attain the lower bound:

$$\tilde C_{\alpha,n}=\frac{\pi}{2}C_{\alpha,n}.$$

This would determine the exact $\tilde\beta$-norm of the Bergman projection and sharpen the estimates in Theorem Mait. The source gives no resolution of the conjecture.