Irreducible characters of the group $S_n$ that are semiproportional on $A_n$

Previously, we dubbed the conjecture that the alternating group An has no semiproportional irreducible characters for any natural $n$ [1]. This conjecture was then shown to be equivalent to the following [3]. Let $\alpha$ and $\beta$ be partitions of a number $n$ such that their corresponding characters $\chi^\alpha$ and $\chi^\beta$ in the group $S_n$ are semiproportional on $A_n$. Then one of the partitions $\alpha$ or $\beta$ is self-associated. Here, we describe all pairs $(\alpha,\beta)$ of partitions satisfying the hypothesis and the conclusion of the latter conjecture.