On the irreducible characters of the groups $S_n$ and $A_n$

Two characters $\varphi$ and $\psi$ of a finite group $G$ are called semiproportional if they are not proportional and there exists a set $M$ in $G$ such that the restrictions of $\varphi$ and $\psi$ to $M$ and $G\setminus M$ are proportional. We obtain a description for all pairs of proportional irreducible characters of symmetric groups. Namely, in Theorem 1 we prove equivalence of the following conditions for a pair $(\varphi,\psi)$ of different irreducible characters of $S_n(n\in\mathbb{N})$:
(1) $\varphi$ and $\psi$ are semiproportional;
(2) $\varphi$ and $\psi$ have the same roots;
(3) $\varphi$ and $\psi$ are associated (i.e., $\psi=\varphi\xi$ where $\xi$ is a linear character of $S_n$ with kernel $A_n$).
Note that (1) and (2) are in general not equivalent for arbitrary finite groups. For the symmetric groups, the equivalence of (1) and (3) validates the following conjecture proven earlier by the author for a number of group classes: semiproportional irreducible characters of a finite group have the same degree.
The alternating groups seem to have no semiproportional irreducible characters. Theorem 2 of this article is a step towards proving this conjecture.