Problems on the maximum of a conformal invariant in the presence of a high degree of symmetry

The problem on the maximum of the conformal invariant
$$2\pi\sum_{k=1}^nM(D_k,a_k)-\frac2{n-1}\prod_{1\leq k<l\leq n}|a_k-a_l|,$$
for all systems of points $\{a_1,\dots,a_n\}$ and all systems $\{D_1,\dots,D_n\}$ of nonoverlapping simply connected domains satisfying the condition $a_k\in D_k$, $k=1,\dots,n$, is investigated. Here $M(D,a)$ is the reduced module of a domain $D$ with respect to a point $a\in D$. It is assumed that $n$ is even and systems of points $a_1,\dots,a_n$ under consideration have a high degree of symmetry.