On irreducible characters of the group $S_n$ that are semiproportional on $A_n$ or $S_n\setminus A_n$. II

In the author's previous paper, the hypothesis that the alternating groups $A_n$ have no pairs of semiproportional irreducible characters is reduced to a hypothesis concerning the problem of describing the pairs of irreducible characters of the symmetric group $S_n$ that are semiproportional on one of the sets $A_n$ or $S_n\setminus A_n$. In this hypothesis, properties of such a pair of characters are expressed in terms of Young's diagrams corresponding to these characters. The theorem proved in this paper allows one to exclude from consideration some stages of the verification of this hypothesis.