The conjecture on subgroup perfect codes in connected Cayley sum graphs of symmetric groups

Let $n\geq 3$, and let a Cayley sum graph of the symmetric group $S_n$ be connected. A subgroup perfect code is a subgroup of $S_n$ that is a perfect code in this graph. Subgroup perfect-code conjecture. Every connected Cayley sum graph of $S_n$ has no subgroup perfect code. This conjecture asserts a generalization of the preceding result for $n=3,4$ or $5$; its status for all $n\geq 3$ is not established in the supplied text.