On a problem of M. A. Lavrent'ev concerning the representability of functions by series of polynomials in the complex domain

M. A. Lavrent'ev has constructed an example of a compact set $E$ in $\mathbb C$ that is the boundary of a domain containing $\infty$ and such that every portion of $E$ separates the plane. Let $\{D_{n_k}\}$ and $\{D_{m_k}\}$ be two subsequences of bounded domains in the complement to $E$ such that every neighbourhood of every point of $E$ contains domains of both subsequences. Let functions $f_1(z)$ and $f_2(z)$ be defined in a disc $U$ that contains $E$. Suppose that they are regular outside $E$, coincide on all Domains $\{D_{m_k}\}$ and are limits everywhere in $U$ of pointwise convergent sequences of polynomials. Are there always domains in $\{D_{m_k}\}$ on which $f_1$ and $f_2$ coincide identically? In this paper we give a negative answer to this question of Lavrent'ev.