The conjecture on subgroup total 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 total perfect code is a subgroup of $S_n$ that is a total perfect code in this graph. Subgroup total-perfect-code conjecture. Every connected Cayley sum graph of $S_n$ has no subgroup total perfect code, except for the graphs in Example S3. The supplied context proves the corresponding assertion for $n=3,4$ or $5$ and identifies the exceptional graphs through the cited example; the extension to all $n\geq 3$ is not established there.