Deformation of plates of small condensers and Belinskii's problem

We study the homeomorphic embeddings of a compact set $K$, a union of nondegenerate continua, into $\overline{\mathbb R}^n$ which preserve the conformal moduli of all condensers whose plates are continua in $K$. Using a result by V. N. Dubinin together with the estimates for the conformal moduli of infinitesimal condensers, we prove that Belinskii's conjecture (that such a mapping extends to a Mobius automorphism of the whole space $\overline{\mathbb R}^n$, corroborated by the author in 1990 for $n=2$ is also valid for $n>2$ if the compact set in question is regular at some collection of $(n+2)$ points. This essentially strengthens the previous result of the author (1992) in which regularity was required at each point of the compact set.