Certain pairs of irreducible characters of the groups $S_n$

The investigation of the pairs of irreducible characters of the symmetric group $S_n$ that have the same set of roots in one of the sets $A_n$ and $S_n\setminus A_n$ is continued. All such pairs of irreducible characters of the group $S_n$ are found in the case when the least of the main diagonal's lengths of the Young diagrams corresponding to these characters does not exceed 2. Some arguments are obtained for the conjecture that alternating groups $A_n$ have no pairs of semiproportional irreducible characters.