On the problem on maximizing the product of powers of conformal radii of nonverlapping domians

A sharp estimate of the product
$$ \prod^4_{k=1}R^{\alpha^2_k}(D_k,b_k) $$
(as usual,$R(D,b)$ denotes the conformal radius of a domian $D$ with respect to a point $b\in D$) in the family of all quadruples of nonoverlapping simply connected domians $\{D_k\}$, $b_k\in D_k$, $k=1,\dots,4$, is obtained. Here, $\{b_1,\dots,b_4\}$ are four arbitrary distinct points on $\overline{\mathbb C}$, $\alpha_1=\alpha_2=1$, $\alpha_3=\alpha_4=\alpha$, and $\alpha$ is an arbitrary positive number. The proof involves the solution of the problem on maximizing a certain conformal invariant, which is related to the problem under consideration.