Asymptotic limit conjecture for Bloch-to-Bergman inclusion norms

Let $\alpha>-1$ and let $C_\alpha(p)$ denote the norm of the inclusion under consideration from the Bloch space into the weighted Bergman space $A^p_\alpha$. The quotient $C_\alpha(p)/p$ measures the asymptotic growth of these inclusion norms.

Asymptotic limit conjecture. For any $\alpha>-1$,

$$\lim_{p\to\infty}\frac{C_\alpha(p)}{p/\big(2e\sqrt{(\alpha+1)(\alpha+2)}\big)}=1.$$

The preceding corollary gives the corresponding lower and upper asymptotic bounds, and the conjecture asserts that the quotient is actually convergent and reaches the upper bound. The result would determine the precise asymptotic growth of the inclusion norms; it is stated as a conjecture in the source.