Prime-graph equality conjecture for intransitive subgroups of alternating groups

Let $G=A_n$ and $H=(S_k\times S_{n-k})\cap G$, where $1<k<n$ and $p\leqslant k$ for every prime number $p\leqslant n$. Prime-graph equality conjecture. If $n\geqslant 12$, then

$$\Gamma(G)=\Gamma(H)$$

if and only if $n$ is odd, $k=n-1$, and $n-4$ is composite. This concerns the unresolved classification of when an alternating group and an associated intransitive maximal subgroup have the same prime graph; the paper notes that the problem is connected with difficult additive questions about representations of integers as sums of distinct primes.