Removing derangements while eliminating transitive subgroups

Determine, for n>5, whether derangements can be removed from \(\operatorname{Der}(n)\) so that the remaining derangements together with the identity contain no transitive subgroup, without removing all derangements mapping a fixed point i to a fixed point j.

Background

The clique-coclique arguments in the paper show that transitive subgroups contained in the remaining connection set preserve the original star-sized independence number. Removing all derangements mapping i to j is one way to prevent this, but it may be unnecessarily destructive.

The question asks for a more economical deletion strategy that eliminates every transitive subgroup from the relevant connection set without deleting an entire point-mapping class.

References

Question 7.4. For n > 5, is there a way to remove derangements from Der(n) so that the resulting set, together with (1), has no transitive subgroups, without removing all the derangements that map i to j?

— A new measure of robustness of Erdős--Ko--Rado Theorems on permutation groups  (2502.14582 - Gunderson et al., 20 Feb 2025) in Question 7.4, Section 7